okay so in this video we're going to have a look at five of the hardest probability questions these tend to involve algebra but may also involve some probability trees and just different methods that we're going to have a look at so we're going to look at five different questions in no particular order of difficulty the one that you can see on the screen is the one we're going to kick things off with so with that being said let's get started okay so looking at our first question now as with all of these i do encourage you to have a go at them so make some notes write the question down see how far you can get and then go through the solution with me so make sure you grab a piece of paper grab a pen and let's get started so it says here there are only r red counters and g green counters in a bag essentially we don't know how many are in the bag a counter is taken at random and the probability that it is green is three over seven so we'll highlight that the counter is put back in the bag and then two more red counters and three more green counters are put in the bag another counter is taken and the probability that it is green is 6 over 13 and it wants us to find out okay well how many red counters and how many green counters were in the bag originally now i think the first first thing that people tend to try and do on this question is because of this language here where it says there are r red counters and g green counters people try and make some sort of algebra using the letter r and the letter g now this can be done although we can actually avoid it and make this question an awful lot easier by just using one letter and thinking about how we can apply that now as with a lot of these questions when we don't know the amount we can write things in ratios or we can create expressions now i'm going to take one approach to this what i think is the easiest approach although if you can obviously apply a different type of method and a different approach that's absolutely fine as long as obviously your answer matches at the end but the method that i'm going to approach so it says here that the probability of getting a green is three over seven now if we write this as a ratio we've got green to red and that's three out of seven so that is three to the remaining number up to seven which is four so we know it's in the ratio three to four now instead of writing three to four we can write that in terms of algebra we can say 3x and 4x obviously it's not necessarily going to be three green counters and four red counters but we have some multiple of three and four in that ratio so instead of writing three to four let's write 3x to 4x there we go and that's our ratio sorted for that first statement it then says the counters are put back in the bag two more red and three more green counters are added so if we use this piece of information here how does that affect our expressions here well let's start with the green as i've got green first in my ratio so my green one would now be 3x the original amount that we don't know but adding in an extra plus three okay exactly three more counters so that'll be my new expression for green and my expression for red would be the four x that we've got at the start plus an additional two counters so there are my new expressions once these new counters have been added in and we know that the green or the probability of picking a green is 6 over 13. so what fraction what expression can we make for the total here for our denominator for our probability well if we add both of these together that's going to give us our total amount of counters in terms of x we have 3x and 4x which adds up to 7x and then we've got 3 add 2 which adds up to 5. so we know there's a total of seven x plus five whatever that value of x is so if we write that this is an equation then looking at that green probability we have the three x plus three as our probability for green out of seven x plus five so let's write that down we have three x plus three over seven x plus five and we have been told that that probability is equal to six over thirteen and there we go now we've got an equation not the nastiest of equations to solve either we just need to do a bit of cross multiplication to get rid of our fractions here so we're going to multiply this denominator up to the 6 and this denominator 13 up to our 3x plus 3. that's going to create a nice straight line equation we'll get 13 brackets 3x plus 3 and that is equal to 6 brackets 7x plus 5 and then we just need to expand that out and solve it so if we expand that out we get 39 x plus 39 times in them both by 13 and that is equal to 42x plus 30. there we go so again removing our smallest x from both sides here just to solve this like a normal equation we will remove 39x from both sides so we've got 39 equals 3x plus 30. now we can subtract 30 from both sides so we get 9 equals 3x and dividing both sides by 3 we'll get x is equal to three now there we go we've solved it we found out what x is but let's not forget we need to actually answer the question here now the question says find the number of red counters and then the number of green counters that were in the bag originally well the information about how many counters were in there originally is in our ratio just here we have 3x and 4x okay that was our original expression that we made for the counters so in terms of the original audio the counters in originally for green we've got three lots of x we know that x is three so let's write this over here we've got green which is going to equal three lots of three which is nine counters and for red we've got four x so four lots of three is 12. and there we go there's our final answer so green is 9 and red is 12. okay not forgetting all we've done is we've subbed into here our value x equals 3. and there we go there's how to solve that okay so there's our first question obviously if you've got your answer 9 or 12 a different method that's absolutely fine there are a couple of different ways of doing this but i think that one there is the most logical or certainly the quickest method in order to solve this one there we go let's have a look at our next question okay so for our second question we're going to have a look at using a bit of a probability tree so it says here there are 10 pens in a box there are x red pens in the box and all other pens are blue jack takes at random two pens from the box find an expression in terms of x for the probability that jack takes one pen of each color and give your answer in its simplest form now again with a question like this we can write our sort of different colors here in a ratio we can say that in terms of our ratio we've got red and we've got blue and for red there we know it's x for blue we don't know at the moment but we know that the total is 10 pence so for blue in order to get the amount of blue we would take the total which is 10 and we take away however many that are red and in this case red is x so we would do 10 minus x and that would be our total there for blue now that's going to allow us to create our fractions here for our different probabilities not different to the last one because we are looking at two separate probabilities here and when we're doing that when we're looking about these sort of journeys it's easy for us to draw a probability tree now this that being said you don't have to draw a probability tree for these but it does sort of make it a little bit more visual and a little bit easier to understand so let's have a look we've got red and we've got blue and on that first pick there when we take that first counter out and just something to mention here although it does say in the words jack takes at random two pens from the box the only way that we can work through this problem is to imagine that we first take one pen out assess the probability of that and then once that pen is removed we take out a second one so we're never going to remove two at once this is we're going to work on the premise that we take one pen and then take the second one so having a look at this then we've got red which is x the probability of red out of 10. for blue our expression there for blue is 10 minus x over 10. again just taking these probabilities from up here x out of 10 and 10 minus x out of 10. now for the second one one of our counters now is going to be removed so if we now take out a red it's going to be different and if we take out our blue again it's going to change this slightly so let's label this up red and blue and this one here will be red and blue now as we are looking at the probability of having different colors that's what it says we want the probability of and let's just highlight it one of each color we don't actually need to look at red red or blue blue but for the purpose of our learning here we'll have a look at filling in all these probabilities so for that first one if we take a red out it's going to be x now minus 1 because we're taking one of those reds out and it's no longer going to be out of 10 counters it's going to be out of 9 sorry pens so x minus 1 over 9. okay so on to our probability for blue if we take out a red to start with our numerator there is not going to change so that is going to stay as 10 minus x but now that is going to be out of 9 as one of those counters is going to be removed so there's our two probabilities there for red and blue on our second part of the journey let's have a look if we've taken out a blue counter so if we take our blue counter the numerator for red is not going to change so it's going to be x and that's going to be out of nine and if we take out one of those counters for blue then the numerator is going to reduce by one so that would be nine minus x and that again would be over nine now again we don't need the full journeys here we just need to have a look at the two different counters so if we go up the red route and down the blue route that's going to be one of our probabilities so we need to multiply those two fractions together and likewise we could go down the blue and then take out a red so let's take these two probabilities and write those out so going up the red and then the blue we have got x over 10 and we need to times that by the 10 minus x over 9. for the other root for blue and then red we've got 10 minus x over 10 and we need to times that by x over 9. so these are the two little fractions that we need to work out so let's get rid of this tree and we're going to figure both of these out so let's get rid of all of that there we go so just looking at these two fractions then now the numerator there we've got x multiplied by the 10 minus x that's going to be 10x minus x squared on the top and that is 10 times 9 which is 90 on the bottom for the second one we've got 10x again minus x squared and that is again over 90. okay so we end up with the two of the same fractions here now let's have a look we need to add these two together just like normal probability trees we're going to have to combine these together so let's just write that out as our actual working so 10x minus x squared over 90 and we're going to add to that the same thing 10x minus x squared over 90. now when we add fractions together i don't know our denominator there isn't going to change so it's still going to be over 90 but we need to combine everything on the top so we've got two lots of the 10x so that is 20x and 2 lots of the negative x squared so minus 2x squared and that is all over 90 and there's our final fraction it does however say here give your answer in its simplest form and if we have a look we can divide the top and the bottom by two so if we do that if we divide everything by two we'll have 10x again on the top minus one x squared and we divide everything by two and then also 45 on the bottom dividing that 90 by 2. and there we go there's our final answer 10x minus x squared all over 45. cool so that's our second question if any of that was a little bit too quick for you again i do have lots of videos on probability and it's particularly these ones involving algebra so we'll link those in the description as well so do make you check those make sure you check those out if any of these are a little bit too quick for you but as i said these are difficult questions so they do take a little bit of time let's have a look at our next question okay so on to our next question now this one is a large question so i would make sure that you write down this question before we get started because chances are i'm going to have to completely remove everything off the screen to fit this all on one page so let's have a look it says there are only green pens and blue pens in a box there are three more blue pens than green pens and there are there are more than 12 pens in the box simon is going to take at random two pens from the box and again remembering what i said on the last question that if we remove two pens we're going to remove one and then the second and it says the probability that simon will take two pens of the same color as 27 over 55 work out the number of green pens in the box now do make sure that you write this question down because i am going to remove this from the screen in a second but here we go let's have a look there are only green and blue and there are three more blue than green there are more than 12 which is what we're going to have a look at towards the end and the probability that someone will take two pens of the same color is 27 over 55 and we're going to work out the number of green pens in the box so like we've done previous ones let's write a little ratio here for blue to green now we've got blue to green and it says here that there are three more blue than green so we could write this in different ways we could write that blue is x and then green would be three less so x minus three or we could write green as being x and then blue as being x plus three as there are three more i think that's probably the easier way to write it here just to avoid having any more negatives than we need so there we go we've got x plus 3 and x and our total again for that is 2x plus 3. there we go so what we're going to do is we're going to create another probability tree again but we are going to be looking at the probability that simon will take two pens of the same color and that has actually been given to us as 27 over 55. so if we have a look at creating this let's see how far we can get on this screen we can have the probability of blue and the probability of green so let's have blue going up and green going down now blue will be x plus three out of the two x plus three so we can write that probability in x plus three over two x plus three and the probability of green there is going to be x over the two x plus three now again we are only really concerned here about taking the same colors so we don't need to fill in every single branch this time although feel free to do that when you are practicing because it is good to be able to create these expressions but if we take out a blue so our numerator is going to reduce the plus 3 is going to go down to plus 2 and the denominator the plus 3 is going to go down to plus 2. so that's going to be x plus 2 over and then it's going to be 2x plus 2. so there we go our numerator has gone down because we look at the same color this time we don't need to do the green one so let's mix that out and go down the bottom root again we've done this root now let's have a look at the other root for same colors which is going to be green green so our green numerator is going to reduce so it's going to go down from x at the moment and that's going to go down by 1 so x minus 1 and our denominator is going to go down to the plus 2 in that position as well so we've got x minus 1 over and that is going to be 2x plus 2. okay so what do we need to do now we need to multiply both of these fractions together so in order to do this let's just create a little bit more space okay so here's all the information that we had written down now we just need to work out the probability of each root so going up the blue blue root we have x plus 3 over two x plus three i'm gonna multiply that together with the x plus two over the two x plus two so this root here and i'm gonna skip a few steps here because i'm gonna multiply the numerators and multiply the denominators but i'm not going to do any working out for that i'm just going to do the double brackets in my head although i would suggest that you write that down but i am going to assume if you're looking at this topic that you're quite comfortable with doing a double bracket expansion so obviously you do your working out for this one but on the top there x plus three times x plus two on the numerators gives me x squared plus five x plus six i'm on the denominator there we've got two x plus three times 2x plus 2 so that's going to be 4x squared then we've got plus 4x and 6x so that makes plus 10x and then 2 times 3 makes 6 so plus 6. and there's our fraction for that probability there we've also got green green so again we've not got a double bracket on the top there we've got x times x minus 1 which is going to give us x squared minus x and again our denominator is going to be the same so 4x squared plus 10x plus 6. now we need to add these two probabilities together okay because the combination of those two probabilities will give us the probability of getting the same colors so if we add those two together and let's write this down the bottom we've got the x squared at the x squared which makes 2x squared we've got a 5x and adding in a minus x so that's going to become 4x okay and again if i just highlight that little bit there that was the 5x and this minus x i've added together to make this plus 4x and also at the start there we have the x squared and the x squared which has made this 2x squared then we've just got the plus 6 at the end and no numbers on the numerator for that bottom one so that's going to stay as plus 6 and again our denominators just stay the same so 4x squared plus 10x plus 6. now if you remember and hopefully have this written down it said in the question that this was equal to 27 over 55 so that is equal to 27 over 55 and now we have our equation to solve and again this isn't an easy one we now have to cross multiply we've got large numbers in the fractions but we need to make this a single line and then we're going to probably have to either look at factorizing or if it doesn't we could potentially have to look at the quadratic formula but we'll see what happens as we work our way through this question but we're at a point here where we've got the majority of the marks on a question like this at this point we've got around three marks out of what this was which was a six mark question so around about the halfway point we now just need to get it into a position where it's a single line and it's equal to zero as we do with all our quadratics and then we'll be picking up our next mark before we look at either factorizing or whatever process we need to solve it but if we get rid of all this tree now we'll just focus on this actual um equation that we have here okay so let's just get rid of all of that and we'll just focus on this last little element here and hope that we get a nice quadratic out of it let's just move that up there there we go okay so we're going to cross multiply now it's not going to be too bad we have got some big numbers involved here though so let's go through that process so the 55 is going up to the 2x squared so we're going to have 55 lots of 2x squared plus 4x plus 6 and that's going to equal 27 lots of the 4x squared plus 10x plus 6. right so expanding these out so times in the left numbers by 55. again you can use a calculator for this one but let's try and do as much as we can so 55 times 2 is 110 x squared 55 times 4 is going to give us 220x and then 55 times 6 gives us plus not too bad times in the left side by 27 so 27 times 4 gives us 108 so we've got 108 x squared 27 times 10 is easy enough plus 27 sorry 270x i say it's easy then almost get it wrong and then we have 6 times 27 which is 162 so plus 162. there we go we've got our straight line we now as i said we need to make this equal zero so everything on the right there let's move that over to the left and see what we get so let's minus the 108 x from both sides to start with so let's minus this so take that away from the 110x squared leaves us with 2x squared then we've got the next one so let's have a look at that number there we've got 270x and we need to remove that and we've got 220x on the left so 220 take away 270 leaves us with minus 50x and then the number at the end 162 330 take away 162 from both sides will leave us with 168 so plus 168 and that now equals zero and that's where you pick up your fourth mark on this question right so next thing let's have a look we want to factorize this so let's see if we can make it a little bit simpler which we can because all of our numbers here are even so we can definitely divide them all by two and as it's zero on the on the right there you're allowed to do that because zero divided by two on the right is still 0. so that now leaves us with x squared minus 25 x plus half 268 which is 84 and that now equals zero and that's potentially going to make it easier for us to factorize now let's see if we can factorize this so 84 at the end and again when you are factorizing always just write down your factor pairs so for 84 we can have 1 and 84 we could have 2 and 42. we can have does it divide by 3 84 i believe just divide by 3 those digits add up to 12 so 84 divided by 3 is 28 so we could have 3 and 28 definitely divide by 4 before and 21 and actually at that point there we can spot that that is going to allow us to make this 25 in the middle so it's definitely going to factorize didn't take too long to get there so if we put that into a double bracket which we can do up here we would have x let's have a think minus 21 they're both negative so x minus 21 and x minus 4 equals 0. that gives us two solutions so we've got x is going to be positive 21 or x equals 4. now if we refer back to some of the language in the question it said that there were more than 12 counters in the bag so chances are it's going to be this x equals 21. um obviously we are going to be looking at the bigger one if there is if it's stated it's going to be more than 12. and let's just refer back to um our original expressions here if blue was x plus 3 or if if we look at the two x plus three two times four would be eight plus three is eleven so that's why this question has stated that there's going to be more than twelve because obviously using x equals four would only give us eleven counters but the actual question asked us to work out the amount of green counters and as you can see green was our letter x there we've got the x equals 21. so in terms of the amount of green counters green equals 21. there we go and there's our final answer if we wanted the amount of blue don't forget it said that there were three more blue counters in green so that there would be 24 blue counters so there we go there's our final answer lots of working out a very big question there a six marker definitely up there with your real top-end questions so there we go there's our question three i believe let's have a look at our next one okay so moving on to our fourth question we've got here that there are white black socks and five white socks in a drawer josh takes a random two socks from the drawer the probability that just takes one white sock and one black sock is 6 over 11. show that 3y squared minus 28y plus 60 equals 0 and then also find the probability that josh takes two black socks so we've been given the probability here for one specific root and that is whereby he takes one white and one black in no particular order so if we look at creating our tree for this or our expressions to start with it's going to be very similar to our last question so if you're okay with the last one this one is very very similar so let's have a look if we create our ratio this time we have black and we have white and we've been given that there are y black socks so y for black and there are also a five there for white and again just like we've done before we'll get the total there and the total for those two is y plus five and there we go so if we go about creating our tree this time we're going to use obviously these expressions that we have here so let's have a look we've got black and we've got white so if we have black going up and white going down so black here is going to be y out of the y plus 5 and white is going to be five over the y plus five now this time we don't need to worry about the black root going up because we want to have those different colors one of each so we're going to go down the white root and then if we went down the white we're going to go up the black root okay so if we take out one of the white counters um in fact let's go upwards to start with looking at this root let's take out a black counter and then a white counter so the white numerator will have not changed so it's still going to be five but it's no longer going to be over y plus five it's going to be over y plus four and there's our second root for that one now let's think about if we took out the white counter to start with and we then take out the black well the black numerator from the start won't change so that will be y over y plus four there we go so there are our two fractions and we're going to take exactly the same approach as we did last time we're going to multiply these together so again just like before let's create a little bit more space okay so now we've got a bit more space we just need to work out these two multiplications of these fractions so for the top one here taking out a black and then a white we've got the numerators so 5 times y is just 5y and on the bottom now we've got a double bracket again which i'm going to skip the working out we get y squared plus 4y plus 5y which is plus 9y and then 4 times 5 which is 20. the same for the bottom root we've got 5y over y squared plus 9y plus 20. so we get the same answer for both of those now we just need to add these two together and if you remember from the question it said that this probability or these roots were equal to 6 over 11. so if we add those two together let's put a plus sign over here we get 10y over that denominator y squared plus 9y plus 20 and that is equal to 6 over 11. and again we've got a similar question we just need to cross multiply so if we cross multiply these and see what we get 11 times 10 on the left there gives us 110 y and that is going to equal and we're going to times this other one by six so we will have six y squared plus six times nine which is 54 y plus six times 20 which is 120. there we go and we have got our quadratic now again we need to make it equal to zero so rather than taking away uh the six y squared we're going to take away the 110y from both sides so we've got y on the right and we need to take away 110 y from both sides that leaves us with negative 56. so we'll have 0 equals 6y squared minus y plus 120 and again we need to go about actually factorizing this so can we simplify it at all first yes we can we can divide both sides by 2 just to simplify it down so we'll have 0 equals 3y squared minus 28y plus 60. okay so we still need to go about factorizing this so let's get rid of some of this working out here and then let's have a go at actually factorizing it so we've got our factors over there which is 60. now let's write the brackets to start with we know it's going to be a 3y and a y and we'll have a look at the factors so for 60 what could we have we could have 1 and 60 2 and 30. bearing in mind one of these is going to be multiplied by three we can have three and twenty is that going to work no that's gonna make 29 do either any of these work so far uh it doesn't look like it what else could we have then we could have four and 15 does that work that could make 12 and 15 that would make 27 so now that doesn't work we could have 6 and 10 that would make 18 and 10 that would work that makes 28 there we go there is our factor pair so 6 and 10. obviously bearing in mind if you have a calculator for this and you could just straight away use the quadratic formula but i'm going to be doing this as a non-calculator method so there we go 6 and 10 we want the 6 to be tripled so we'll put the 6 in the other bracket the 10 over here we're trying to make minus 28 in the middle and positive 60 at the end so they are both going to be negative and that's going to give us our two solutions there we've got y is equal to 10 over 3 and y is equal to 6. right there we go so we know that y cannot be a fraction because it is the amount of blue counters um that we're looking at here i think it was about counters i've actually now forgotten about what the question was about because it was a so long ago so always make sure that you do write these questions down so this particular question was about socks so it's about black and white socks not counters so the answer for y that we're looking for we've got y equals six so we can put that into our original ratio up here y equals six and um we have actually shown the first part of the question i actually just went completely past part a which was to show that 3y squared minus 28y plus 60 equals zero and part b was to find the probability of taking two black socks from the drawer so in order to order to work out the property of two black socks we now know that there are six black socks out of the total which is eleven so it is going to be 6 over 11 for the probability of taking the one black sock and then we need to find the probability of getting the two black socks so we will have a 6 over 11 and that will be multiplied by 5 over 10 which is equal to 30 over 110 and that can actually simplify further let's just get rid of some of this we can divide the top and bottom by 10 which would give us 3 over 11 as our final answer there okay cool so we proved our quadratic we uh solved that in order to find how many or what the value of y was so we knew that how many black socks there were and then once we knew that there were six we know that there's 11 in total so it was 6 over 11 multiplied by 5 over 10 once 1 has been removed that gives us 30 over 110 and that simplifies to 3 over 11 and that is the end of that question there that was actually a 7 marker and an awful lot of working out although i don't think the algebra was quite as complicated in that one as it was in the previous one but there is an awful lot of working out in that question so there we go there was question four our final answers there let's have a look at our final question okay and for our final question then and you're gonna see there's only one real difference or major difference in this question and we're going to apply that in the first step and this question is a calculator question so that might give you a little hint as to what we're going to have to do at the end now it says here there are some red counters and some white counters in a bag and let's quickly make our ratio straight away at the seven at the start sorry seven of the counters are red and the rest are white so seven are red we don't have a number here or a letter to use so we can pick our own letter let's go with the letter x so in order to get the amount of white counters x is our total let's say white would be x minus seven or we could actually put white as being x and then we could have our total uh in a slightly different way so there are different ways of doing this because we could have our total now as being the x value or we could rearrange this and give white as the x value and then make our total there as x plus seven which is probably a little bit easier to do but when you're not given um the letter for any particular part there are two different ways that you can create your expressions both will work but this is probably our easier way to do this so for this one then it says alfie takes that random a counter from the bag he does not put it back and then he takes another one the probability that the first counter is white and the second counter is red is 21 over 80. now the nice part about this question is that only gives us one route to work out so the first counter is white the second counter is red and our probability there is 21 over 80. so actually this hopefully shouldn't we could be too much of a large tree because we're only going to look at that one root so if we draw it in we have got two options that's going to be red and white and we're going to take the first counter being white now the first counter there is going to be x out of the total which we've written down as x plus 7 again x over x plus 7 just taken from our ratio if we were to write red in as well which we don't need to but it would be 7 over x plus 7 for red now the second council we are going to take is going to be red so that's going to be the uproot i'm not going to draw the whole tree for this one because we're only going to look at this one part of the journey which is taking out a white and then taking out this red now the probability and this is why i wrote down the probability for red there because we've taken out a white the numerator is not going to change so that is still going to be seven although it is going to be out of x plus six now as one of the counters has been removed and not replaced so there's our two fractions x over x plus seven we need to multiply that by this seven over x plus six and those two probabilities there are going to equal 21 over 80. let's see if we can fit this all in so x times 7 will give us 7x on the top and then we are going to have a quadratic there x plus 7 times x plus 6 which is going to give us x squared plus 6x and 7x which is going to be 13x plus 6 times 7 which is 42. and just about fit that in now if we get rid of the tree we can set that equal to our fraction and go about forming our quadratic so let's get rid of all of this and again you can hopefully see a lot of similarity between this and some of our previous questions so we have 7x over x squared plus 13 x plus 42 and that probability is equal to 21 over the 80. right so again just like before we need to cross multiply here so we're going to times the 7x by 80 and this quadratic here by 21. so let's write that out so we have 80 times the 7x and that is going to equal multiplied by the x squared plus 13 x plus 42 and now if we expand that obviously we just need to multiply them out so 80 times 7x is going to give us 560x that's going to equal 21x squared plus we've got 13 times 21 which is 273 as i said you do have a calculator for this one as well 273 x plus 42 times the 21 which is 882 so plus 882 and you can see why you have a calculator for this one and there we go now we need to make that equal zero so we need to take away this 560x from both sides so we'll get zero equals 21 x squared and let's have a look 273 take away that 560 leaves us with negative 287 so minus 287 x plus the 882 there we go right so we have a pretty nasty looking quadratic there and i wouldn't know off the top of my head whether they will simplify i suppose you could test them if you have a calculator there are little tricks to see if things do definitely divide but as we have a calculator we don't actually need to okay we don't need to try and factorize this even if it does factorize which we know it has to factorize because there is x while the value of x is an amount of counters so we must be able to factorize it to get a whole number but there is absolutely no point in us sitting and trying to factorize that which could take us a little bit of time we may as well just plug it straight into the quadratic formula so hopefully we know our quadratic formula minus b plus and minus the square root of b squared minus 4ac and that is all over 2a now for this one our values we have a equals 21. we've got b is equal to negative 287 and we've got c is equal to 882. so what we need to do is plug these values into the quadratic formula and find our value of x so let's get rid of all of this and let's just go about typing this in so we've got minus the b which is going to turn that negative 287 into 287 plus and minus the square root of b squared that's negative so we'll put minus 287 in brackets squared obviously really square it's going to become positive anyway so we could just put the positive version in take away 4 times 21 times the 800 882 all over two are two lots of a so two times 21. now we're going to get the one where we do the positive and the one where we do the negative let's type them both in so fraction 287 plus the square root of 287 squared take away 4 times 21 times 882 all over 2 times 21 and we get the answer nine for that one let's go back in and put the minus sign in and see what we get for that there we go and we get the answer 14 over so look 14 over three there we go 14 over 3 so we know that our answer has to be the 9 we can't have 14 over 3 counters here we have to have a whole number so we know that that value of x and let's label that back at the top here must be nine white counters let's just read the question it says work out the number of white counters in the back at the start so there's our answer nine white counters okay so we can write that down here number of white counters equals nine there we go right okay so again another little topic being involved there using the quadratic formula but as i said that one there potentially well will have factorized but would have taken you a lot longer so if you do have a calculator and you are able to use the quadratic formula then you may as well just use it on a question like that but there we go that was our final question hopefully that gave you a bit of an insight there into some really tricky questions to do with probability as i said though a lot of these i have done little videos on so do check those out in the description if any of these were sort of you know quite tricky or something you feel that you need to work on but there we go hopefully you liked the video if you did please like please comment please subscribe and i'll see you for the next one [Music] [Music] oh you