basic probability so when we think of basic probability um there are some terms that we're going to talk about in actually on this page but the first thing you have to understand is sample space so spinner has seven equally spaced numbers list the sample space in the event of getting an odd number on the spin sample space is literally every single option you could possibly get so if I were to tell you to list the sample space that's actually going to be set theory so you would say well the sample space or however you want to label it that's often labeled with like a capital letter the potential options you have look like this right that is your sample space this is vitally important when you talk about statistics the reason I say that is and I don't even if we get into it in here but if you get into higher probability they start talking about the law of large numbers because sample spaces can skew results clearly for example if I were to survey this particular class on one question it's going to look completely different than if I survey the fifth grade class on that same exact question right because y'all are all the same age you're like we have just narrowed down our sample space and so often and this is why you have to really pay attention to statistics that come up in like a news story like I am the most critical person when they start handing me numbers because I want to know where the numbers came from what the sample space is and why are they telling it to me in a specific way um often news stories will specifically give you certain numbers because it sounds the way they want it to sound all right and so sample space is one of the things you want to evaluate when you're being told is specifically statistics because sample space changes my probability of something happening when there's only seven much better than if they're 700 right and so knowing that sample space is important um the event of getting an odd number this is also going to change based on sample space in this case my probability is going to be one two three four four out of seven um probability is sometimes done in fraction form sometimes done in percentage form so when you're doing like a standardized test you want to pay attention to what the answers look like and make sure that you recognize your answer based on what they're asking for theoretical or classic classical probability the ratio of a number of favorable outcomes to the number of equally likely possible outcomes experimental probability is the ratio number of times the event occurs compared to the number of Trials I'll give you an example theoretical probability for flipping a coin is one half okay if you actually to do that in an experiment you have to do it probably over a hundred times to get it close to 50 percent it just I mean you may hit it at four but um practically if you flip a coin to get that 50 consistently you're gonna have to get a large example or a large experimental to do it and so you have theoretical the actual probability of flipping a coin is 50 because there are two options and if you want to have this one out of two but if you were to do experiments or sometimes you don't have that concrete what's going on they actually have to do experiments and see what is my what keeps happening what is my stat so a lot of times a good example is sports right there is no um theoretical probability really for number of free throws that you get percentage of free throws that is an individual basis that's based on skill that's based on uh practice that's based on lots of things so that is going to be an experimental probability based on an individual player their percentage of free throws is really personal to them right now you can have a team percentage and all that but again that is purely experimental that is based on what they have actually done how many times they have shot it how many times they have made it and so these are the two terms that you're going to see come up when you do probabilities theoretical is it concrete we know what my options are and we know what I'm looking for or is it experimental where we have to see what historically they have done to be able to say what the percentage is there you'll also see experimental probability when you see like error ratios in like manufacturing all right they're going to put those in place and they're going to say we don't want our error ratio to be outside of this and then they're going to watch it experimentally to make sure that they're staying Within what they want all right all right assuming that we have a non-weighted six-sided die um table displays the results of her 10 rolls of a die State the theoretical and experimental probability for each event rolling a four there's only one four on one die there's six sides um theoretical that is going to be one out of six that's what theoretical would be right so um if I am doing theoretical probability I'm just going to say it's going to be one out of six those are that's there's one four there's six sides right um rolling an odd number well um rolling a um odd number well there's three odd numbers and there's only six right so three out of six also known as one out of one out of two um they then give us a chart right and so the chart actually tells us her roles okay um and the frequency and what they did as far as those roles go this then is what she actually did this is what she did she rolled it um here's the numbers one two three four five six right and then how many times she um got those numbers so rolling a four then right here she actually rolled it three times out of how many rolls ten so the experimental actually ended up looking like this right um we went from about 16 to like 30 percent right um rolling an odd number well we have zero here we have one here we have two here right um this one actually ended up um also being three out of ten right and so this went from like a fifty percent to like a thirty percent when she did experimental and so this is where when I talk about law of large numbers she would have to roll it more than just 10 times to get close think the more experimental you do when you have a theoretical that's concrete Like a Rolling of a die is the closer you're gonna get to the theoretical so the the more experiments you do the closer you're going to get to the theoretical probability just like flipping a coin the more times you flip a coin the more your percentage is going to hone in on 50 if you're looking for heads or looking for tails and so that's what's going to happen if she had done more than 10 you'd probably see those creep closer to each other um because that's the probability the theoretical is going to usually win out if you do enough experiments theoretical and experimental so theoretical is just what you're looking for over how many options you have experimental is the number of times it actually happens when you're trying it divided by how many times you tried it um an event so your probability is going to go from zero to one those are your options it's a zero percent possibility or a hundred percent possibility or everything in between so probability is not going to go greater than 100 it's not going to go below zero percent all right so zero to one is always going to be your probability which means complement complement is a term that we used in um set theory what does complement mean complement means the opposite of what you're looking for so if I'm looking for heads the complement of heads would be the tail probability which yes always adds up to one the head's probability is one half so then the complement of that would also be one-half they must always equal one so if you have a probability of one thing and you're looking for the complement you can just do the probability of the one like it shows here and subtract it and that's going to give you the complement of it same thing with like a 90 degree When you subtract from 90 you get the complement of the angle um a certain event if you know for sure that something will happen the probability is going to be one impossible events if you know for sure that it is not possible for something to happen your probability is going to be the empty set which is zero and so those are just some terms that you're going to look for I know how many births I had I know how many births were um multiples and I want to find ones that weren't there's two ways to do this you can go ahead and subtract and put that over your total or you can find the percentage of the multiple births and subtract from one so I can either just subtract at the beginning or I can say well one minus my multiples that I have here over how many I had right and then that complement when I round percentages is going to look like that because if I calculate this it's going to be about three and a half percent and so if if I have been given information and then they ask for the exact opposite you can just take that and subtract from one they're always going to add up always so like if I have independent events let's say I'm rolling a blue die and I'm rolling a red die those don't affect each other right they're whether I roll one first with the other first like the probabilities are not going to change I'm not taking a number off the die I'm just going to multiply those um where you find that you have um situations that affect each other for example if I have a box of chocolates and I'm you know the kind that wants only certain ones and I take out one and it's a caramel um and I'm like okay I'll eat this one right my probability for the second time I eat it eat a chocolate just went down because I'm not replacing it right um and so there are times when whatever happens the first time affects the second time and so you're still going to multiply those probabilities but then you have to make the second time contingent on what happened the first time very often in theoretical like say they want you to choose one and then choose the next they're assuming that your first choice was successful okay and so they may not always tell you that but if you're choosing from marbles in a bag and you say what's the probability of choosing a red when not replacing it and then choosing a blue one you assume that the red one was successful so that you did pull out a red one right because number of Reds and number of marbles just went down and so oftentimes that second one assumes a success on the first try and they may not tell you that in the problem that they assume a success you still multiply them you just have to consider what happened so mutually exclusive mutually inclusive so all right um if A and B can both happen they are inclusive um if they cannot both happen they are exclusive all right so um for example you will not have calculus and pre-calculus students that cross over those are mutually exclusive mostly because precalculus is a prerequisite right however I I could have geometry in Algebra 2 crossover like those are not technically exclusive events right um and so you have certain times when um it's going to be mutually exclusive meaning one cannot happen along with the other it's one or the other and then you have it where you could have one or the other or both um so for example you could have someone who um plays baseball and who plays basketball those are not mutually exclusive events right um so you um those could cross and when you have those you have to consider the probability of this the probability of that and then you have to subtract out the Crossovers because you've counted them twice okay so if I were to take um the baseball team and compare it to the basketball team I would have to make sure that once I did that I took out the Crossovers because if not I've counted somebody twice in there if there's a crossover right and so same thing here I do not ninth grade and pre-cal are mutually exclusive events right however in high school those are the only mutually exclusive events if I said I want to look at all my seniors and all my pre-cal students I have now counted some people twice and I have to take out the senior precal students one time because I've already counted senior pre-cal students right um and so that's where you look at mutually exclusive they don't they don't affect each other at all there's no way there's a crossover or they're not mutually exclusive we have some crossover that we have to consider or we will count them twice and it will be wrong they use a Venn diagram here because this is a good picture of it um if you have uh in your brain if you start looking at these questions on like a standardized test they like to throw out like a million different numbers at you and you literally have to organize it in your brain and a good way to organize it's a Venn diagram and just see okay what am I double counting here and that gives you a picture of that where they cross over has been double counted at that point and so if you look at and um A and B are mutually exclusive um it's basically um exclusive you're just going to add the two probabilities if it's inclusive you're going to add the two probabilities and then you're going to take out this guy right here A and B so this is really the only difference between inclusive exclusive you still add up the probabilities of both events happening but then you take out the and you take out where it crosses over and so you're going to go back to like your set theory what does the and mean take that out and so if you look at example seven a survey of 371 teens shows that 266 own a dog and 157 own a cat if 108 own both a dog and a cat what is the probability that a randomly selected teen owns either a dog or a cat all right so these are not mutually exclusive events right clearly they said there's somebody that does both right um and so we're going to take the probability of the dog we're going to add to that the probability of the cat and then we're going to subtract out the probability of the dog um and the cat right and then we've got our correct probability now probability is not just the 266 for dog that's going to be 266 out of all the people that they looked at then we have the probability of a cat and then we're going to subtract out the probability of both what's nice about these is I have a common denominator so I'm not reducing anything till the end obviously um when you do all of that you end up with something like this again most of the time probability is going to be in percentage which means divide and multiply by a hundred so approximately 85 percent is one or the other the rate they've given us a table uh summarize the results of a survey they ask 100 434 students how many books they read last summer you want me to survey that in here is anybody at nine plus I just need to know if you read nine books over the summer uh so how many books they read last summer determine the probability that a randomly selected student read each number of books for or fewer so um four or fewer we have um zero one to two and three to four these are um not gonna have to really subtract anything else because they're mutually exclusive if you read three or four we're not going to assume they counted you in the zeros right I mean like that just doesn't make sense right um and so for this one um we can actually just add up our probabilities we can say well if it is four or fewer less than or equal to four then we can say just the probability that they had zero the probability that they were in the one to two group and the probability that they were in the three to four group and we're good to go right and so we can just say 3.9 percent um by the way Technically when you're adding these technically you're gonna see it written like that it's gonna work out the same but technically you're gonna see them write it in decimal form because you don't really add percentages you add decimals but it's going to work out the same when you do it you're going to get approximately 37.8 percent assuming that those were not rounded 37.8 percent assuming I've already done a how do you think I'm going to do fiber more I'm just going to do a compliment right because if it's four or less five or more I'm going to say uh well I already calculated him I'm just gonna do that right if I had not calculated a I could do it either way I would likely just add five to six seven eight and nine right but if I've already done a I'm just going to do complement of a right make sense so don't do more work than you have to all right