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Main content
Current time:0:00Total duration:7:31
UNC‑2 (EU)
UNC‑2.A (LO)
UNC‑2.A.4 (EK)
UNC‑2.A.5 (EK)
UNC‑2.A.6 (EK)

Video transcript

so we're told that Amanda young wants to win some prizes a cereal company is giving away a prize in each box of cereal and they advertise collect all six prizes each box of cereal has one prize and each prize is equally likely to appear in any given box Amanda wonders how many boxes it takes on average to get all six prizes so there's several ways to approach this for Amanda she could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect on average to get all six prizes or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes to take to win all six prizes so for example she could say all right each box is going to have one of six prizes so there could be she could assign a number for each of the prizes one two three four five six and then she could have a computer generate a random string of numbers maybe something that looks like this and the general method she could start at the left here and each new number she gets she can say hey this is like getting a cereal box and then it's going to tell me which prize I got so she starts her first experiment she'll start here at the left and so say okay the first cereal box of this experiment of this simulation I got prize number one and she'll keep going the next one she gets prize number five then the third one she gets prize number six then the fourth one she gets prize number six again and she will keep going until she gets all six prizes you might say well look their numbers here that aren't one through six there's zero there's seven there's eight or nine well for those numbers she could just ignore them she could just pretend like they aren't there and just keep going past them so why don't you pause this video and do it for the first experiment on this first experiment using these numbers assuming that this is the FIR box that you are getting in your simulation how many boxes would you need in order to get all six prizes so let's let me make a table here so this is the experiment and then in the second column I'm going to say number of boxes number of boxes you would have to get in that simulation so maybe I'll do the first one in this blue color so we're in the first simulation so one box we got the one and actually maybe I'll check things off so we have to get a 1 a 2 a 3 a 4 a 5 and a 6 so let's see we have a 1 I'll check that off we have a 5 I'll check that off we get a 6 I'll check that off well the next box we got another 6 we've already have that prize we're going to keep getting boxes then the next box we get a 2 then the next box we get a 4 then the next box the numbers of 7 so we will just ignore this right over here the box after that we get a 6 but we already have that prize then we ignore the next box is 0 that doesn't give us a prize we assume that that doesn't that didn't even happen and then we would go to the number 3 which is the last prize that we need so how many boxes did we have to go through well we would only count the valid one the ones that gave a valid prize between the numbers 1 through 6 including 1 and 6 so let's see we went through 1 2 3 4 5 6 7 8 boxes in the first experiment so experiment number one it took us eight boxes to get all six prizes let's do another experiment because this doesn't tell us that on average she would expect eight boxes just just meant that on this first experiment it took eight boxes if you wanted to figure out on average you want to do many experiments and the more experiments you do the better that that average is going to the more likely that your average is going to predict what it actually takes on average to get all six prizes so now let's do our second experiment in a river it's important these are truly random numbers and so we will now start at the first valid number so we have a two so this is our second experiment we got it - we got a 1 we can ignore this 8 then we get a 2 again we've already have that prize ignore the 9 5 that's the prize we need in this experiment 9 we can ignore and then 4 haven't gotten that prize yet in this experiment 3 haven't gotten that prize yet in this experiment 1 we already got that prize 3 we already got that prize three already got fries - 2 already got those prizes 0 we actually got all of these prizes over here we can ignore the 0 already got that prize and finally we get prize number 6 so how many boxes did we need in that second experiment let's see 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 boxes so an experiment - I needed 17 or Amanda needed 17 boxes and she can keep going let's do this one more time this is strangely fun so experiment 3 remember we only want to look at the valid numbers we'll ignore the invalid numbers the ones that don't give us a valid prize number so 4 we get that prize these are all invalid in fact and then we go to 5 we get that prize 5 we already have it we get the two prize 7 and 8 are invalid sevens and valid 6 we get that prize sevens invalid 1 we got that prize 1 we already got it 9s invalid - we already got it 9 is invalid 1 we already got the one prize and then finally we get prize number three which was the missing prize so how many boxes valid boxes do we have did we have to go through let's see 1 2 3 4 5 6 7 8 9 10 10 so it's only 3 experiments what was our average well with these three experiments our average is going to be 8 plus 17 plus 10 over 3 so let's see this is 25 35 over 3 which is equal to 11 and 2/3 now do we know that this is the true theoretical expected number of boxes that you would need to get no we don't know that but the more experiments we run the closer our average is likely get to the true theoretical average
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