[Music] in 5.2 we're going to continue our exploration of probability and define this new word uh odds and we'll talk about how you calculate odds for and odds against if you know the probability or vice versa so let's just start with this example suppose that at the beginning of a regular season the riders are given a 25 chance of winning the gray cup so let's define some of the terminology that we learned uh in 5.1 and then we'll continue that to discuss odds so the event in this situation uh the riders are given a 25 chance of winning the grey cup i think the event is them winning and what is the probability that this event occurs well they tell me that in the question the probability of event a is .25 or 25 or i could write that as one quarter what about the complement of this event so remember back in set theory we defined the word complement if you're not sure about those the terminology maybe go back to section 3.1 and just refresh your memory there but the complement of the event is is a not happening okay so it's not a it would be a prime that was the notation that we used for complement or some textbooks use a bar as well but in this textbook we use a prime and so the complement of event a is that the riders don't win the gray cup so the probability of the complement so that's probability of a prime well if there was a one quarter chance that they would win the grey cup then there's three quarters chance that they won't win the grey cup so then when we're talking about odds odds in favor or odds against so you might have heard this uh listed as in oh that team has a four to one odds of winning or three to one odds of winning odds in favor and odds against are a ratio and here's how you calculate odds in favor you take the probability of event a and then you divide it by the probability of event a not occurring so the probability of event a here was one quarter the probability of a not occurring was three-quarters and i really don't like this two-tiered fraction so i'm going to multiply the top and bottom by four that allows me to say that four divided by four is one and on the top i just have one and then this four and this four divide out to be equal to one and on the bottom i just have three and so now my odds in favor of the riders winning the grey cup are one to three those are odds in favor odds against would be the opposite so three to one you can see here in this three to one or this one to three the four did not appear anywhere here there was a four right three quarters or here there was a four one quarter when we divided though in order to calculate the odds in favor we divided the two probabilities and so that four that denominator divided out now let's just write some definitions for what we just talked about and then we'll try a few more questions so the complement of event a is another event and it's the event that a will not occur and if i want to calculate the probability of not a i can also do this i can think about it or i can take 1 and subtract the probability of a remember in the writer's question we had a one-quarter chance that they would win so then three quarter chance they wouldn't win so another way of writing that would be one minus a quarter and then i could get three quarters odds are a way to express a level of confidence about an occurrence of event so odds in favor again it's a ratio and odds against is a ratio if this is the probability of a over the probability of not a and we can write it like this so odds against would be probability of not a occurring over probability of a or you could write it like this okay let's try a few more examples janara is holding all the cards from a standard echo playing cards they ask mario to choose a single card without looking so all the hearts are in the in jannar's hand determine the odds in favor of mario choosing a face card so h is going to be all the hearts i just need to organize what's happening here then we've got a bunch of face cards so that's well if it's all the hearts you've got jack of hearts queen of hearts king of hearts any more face cards that are hearts i don't think so so how many hearts are there 13 how many face cards are there three how many non-face cards are there 10 so if i want the odds in favor i need the probability of choosing a face card over the probability of not choosing a face card so the chance of mario choosing the face card is 3 out of a total of 13 cards and not choosing the face card is 10 out of a total of 13 cards so the odds in favor are 3 to 10. so just be careful because this isn't a probability this doesn't say that there's three face cards out of a total of 10 cards that's not what this says so that's why it's it's kind of helpful to write this out as that ratio just so you make that distinction between a probability calculation and an odds calculation so if that's the odds in favor odds against would be 10 to 3. what about this one what are the odds against a randomly chosen day of the week being sunday okay so i need to really identify for myself what are the events here maybe i'll just start with even combining event and probability so the probability of it being sunday well there's only one sunday out of a possibility of seven days so the probability of not sunday of it not being sunday is six out of seven so now what did they ask me for odds against so that would be probability of not to the probability of it actually being sunday this time i'm going to write it just as the ratio uh written sideways like this instead of in that fraction format so not sunday was six over seven to one over seven and when you usually have a ratio you don't mix fractions and ratios together so i'm going to multiply both sides of this by seven just to clean it up a little bit so then my odds against a randomly chosen day of the week being sunday are six to one so there's six opportunities for it to not be sunday and one opportunity for it to be sunday in the same way when we had that grey cup example when we were trying to do odds in favor well there was one opportunity for them to win and three opportunities for them to not win because that odds in favor related to probabilities a quarter and three quarters hopefully you can see how the odds are different than the fraction probabilities it's not like something out of a total number of outcomes in the sample space how about this one a program randomly selects a student's name from a college database to award a hundred dollar gift card gift card for the bookstore odds against the selected student being in first year are 57 to 43 determine the probability oh so now they're giving me the odds and they're asking me for probability so well let's read it again the odds against the person being in first year and then determine the probability that they are in first year so which number here do you think represents the first year if it says the odds against being in first year would that be the 57 or the 43 i think it's the 43. remember that last example that we just talked about when we had odds in favor this was the winning of the grey cup and then when we had odds against winning it was looked like this and so this number came from the win so here if it says odds against being in first year this number the second number must represent the being in first year can you think about how we took these two numbers and how what i would do to these two to get to the four i mean it looks like i would odd but let's just see if the other one worked out too here my odds against were six and one what was my total number of outcomes seven ah what about this here my odds were three and ten and what was my total number of outcomes 13. so it looks like i think i can add them to get the total number of outcomes so let's just do that and now we can calculate the probability of the winner being in first year 43 out of a total of 100 so 43 chance a hockey game has ended in a tie after some overtime so the winner needs to be decided by a shootout coach has to decide whether ellen or britney should go first in the shootout coach wants to use the best scorer first so they have a better chance of winning right away so the decision will be based on the player's shootout records all make sense ellen has had 13 attempts and she had eight goals scored maybe while i'm doing this i'm just gonna say that means that five goals not scored and then britney she's had more attempts and she had 10 goals scored so for britney that means that seven goals were not scored and then who should go first i think in this case i only need to really compare the probabilities i'm gonna say the probability of ellen is 8 out of 13 so that is 0.615 and then the probability of britney scoring would be 10 out of 17 so that's 0.588 i think those numbers really show that ellen should go first now there's another player on the team josie is she's only had three shootout attempts but she scored twice so should the coach put her first so let's just see the probability that josie will score will be two out of three attempts but that's that's actually higher so what do you think would you put josie in i guess it depends on what kind of a risk taker you are maybe it depends on your team's record just by looking at the numbers she's only had three attempts and ellen has had 13 attempts so probably ellen is a little bit more consistent i would not be putting josie in but if you want to take the chance you know that maybe you have some other data other other observational data on josie maybe ellen's not having a great day you may you may want to put josie in so last example we've got some grade 12 students they're holding a charity carnival to support a local animal shelter the students have created a dice game so they have something called bim and then a card game they're calling zap so now we've got some odds so the odds against winning bim are five two so i'm just gonna underline and just make sense of that for a second so odds against winning are 5'2 so the 2 is the number that represents the win and the odds against winning zap are 7-3 so again because it says against winning that would mean that the three represents the win which game should madison play if she wants the best chance of winning so i need to figure out the probability of winning bim and the probability of winning zap and then i'll compare those two probabilities and i'll see which probability is higher the problem is they give us odds maybe what i said previously is a little bit confusing so you can also take this kind of a question and just change it around instead of saying odds against you can change it to say odds in favor of bim are two to five and just flip this around and then odds in favor of zap would be three to seven the total here is seven and the total here is ten so now i can calculate the probabilities probability of winning bim so winning right because it's odds in favor favor would be would be meeting that event so winning so probability of winning bim would be two out of seven so if i divide that i get .286 and the probability of winning zap well this is the number that represents the win so three out of ten so 0.3 so then i look at these two numbers and i see that madison should be playing zap there's a higher probability of winning zap than winning bin so here i just wanted to highlight these definitions odds in favor you start with the probability of the event occurring over the probability of it not occurring and then you flip it for odds against and don't confuse these ratios here with the actual probabilities the total number of outcomes does not appear in the ratio because the total number of outcomes got divided out when we had that two-tier fraction okay so that was it for 5.2 probability and odds and in the next section we'll start incorporating some counting methods set theory and set notation in order to solve more complicated probabilities thanks for watching you