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# Probability models example: frozen yogurt

CCSS.Math:

## Video transcript

let's say that you love frozen yogurt so every day after school you decide to go to the frozen yogurt store at exactly four o'clock 4 o'clock p.m. now because you like frozen yogurt so much you are not a big fan of having to wait in line when you get there you're impatient you want your frozen yogurt immediately and so you decide to conduct a study you want to figure out the probability of there being lines of different sizes when you go to the frozen yogurt store after school exactly at 4 o'clock p.m. so in your study the next 50 times you observe you go to the frozen your crystal at 4 p.m. you make a series of observations you observe the size of the line so let me make two columns here line size is the left column and on the right column let's say this is the number of times observed so x times observed observed all right times observed my handwriting is OB s e e r VD r times observed all right so let's first think about okay so you go and you see a look i see no people in line exactly or you see no people in line exactly 24 times you see you see one person in line exactly 18 times and you see two people in line exactly 8 times and in your 50 visits you don't see more than you never see more than two people in line i guess this is a very efficient cashier at this at this frozen yogurt store so based on this based on what you have observed what would be the what would be your estimate of the probability of finding no people in line one people in line or two people line on at four pm on the days after school that you visit the frozen yogurt store and say you only visit it on weekdays where there are school days so what's the probability of there being no line a one person line or two person line when you visit at 4 p.m. on a school day well all you can do is estimate the true probability the true theoretical probability we don't know what that is but you've done 50 observations here right this is a notice this adds up to 50 18 plus 8 is 26 26 plus 24 is 50 so you've done 50 observations here so you can figure out what are the relative frequencies of having zero people what is the relative frequency of one person on the relative frequency of two people in line and then we can use that as the estimates for the probability so let's do that so probability estimate I'll do it in the next column so probability probability estimate and once again we can do that by looking at the relative frequency the relative frequency of zero well we observe that 24 times out of 50 and so 24 out of 50 is the same thing as zero point four eight or you could even say that this is 48 percent now what's the relative frequency of seeing one person in line well you observe that 18 out of the 50 visits 18 out of the 50 visits that would be a relative frequency 18 divided by 50 0.36 which is 36 percent of your visits and then finally the relative frequency of seeing a two person line that was eight out of the 50 visits and so that is 0.16 and that is equal to 16 percent of the visits and so there's interesting things you remember these are estimates of the probability you're doing this by essentially sampling what the line on 50 different days you don't know it's not going to always be exactly this but it's it's a good estimate you did it 50 times and so based on this you'd say well I'd estimate the probability of having a zero person line is 48 percent I'd estimate that the probability of having a one person line is 36 percent I'd estimate that the probability of having a two person line is 16% or 0.16 and it's important to realize that these are legitimate abilities remember to be a probability it has to be between zero and one it has to be zero and one and if you look at all of the possible events it should add up to one because at least based on your observations these are the possibilities obviously in a real world there might be some kind of crazy thing where more people go in line but at least based on the events that you've seen these three different events and these are the only three that you've observed based on your your observations it's these three should at because these are the only three things you've observed they should add up to one and they do add up to one let's see 36 plus 16 is 52 52 plus 48 they add up to one now if once you do this you might do something interesting you might say okay you know what over over the next over the next two years you plan on visiting 500 times so visit next visiting visiting 500 times so based on your estimates of the probability of having no line of a one person line or a two person line how many times in your next 500 visits would you expect there to be a two person line based on your observation so far well it's reasonable to say well a good estimate of the number of times you'll see a two person line when you visit 500 times well you say well there's going to be 500 times and it's a reasonable expectation based on your estimate of the probability that 0.16 of the time you will you will see you will see a two person line or you could say eight out of every 50 times and so what is this going to be let's see five hundred divided by 50 is just 10 so you would expect you would expect that 80 out of the 500 times you would see a you would see a two person line now to be clear I would be shocked if it's exactly 80 ends up being the case but this is actually a very good expectation based on your observations it is completely possible first of all that your observations were off that you you know that it's just this is just the random chance that you happen to observe this many or this few times that there were two people in line so that could be off but even if even if these are very good estimates its profits possible that's something that you see a two person line 85 out of the 500 times or 65 out of the 500 times all of those things are possible whenever you think and it's always very important to keep in mind you're estimating a - you're estimating the true probability here which you it's very hard to know for sure what the true probability is but you can make estimates based on sampling the line on different days by making these observations by having these experiments so to speak each of these observations you could view as an experiment and then you can use those to set an expectation but none of these things do you know for sure that they're definitely going to be exactly 80 out of the next 500 times