00:01
It's given in this exercise that the probability that a randomly selected box of a certain type of cereal has a particular prize is 0 .18.
00:12
And we're going to purchase box after box until we have obtained two of such prizes.
00:19
So let's define a random variable x as the number of serial boxes until we have obtained two prizes.
00:41
Now here each of the cereal boxes that we purchase can be viewed as a bernoulli trial.
00:45
There's only two outcomes of interest, either has the prize or not.
00:50
And we can assume that their outcomes are independent.
00:55
In a series of bernoulli trials, the number of trials until the first r successes is a negative binomial random variable.
01:05
So here we can say that x is a negative binomial with probability of success 0 .18 and number of successes, the probability mass function for a negative binomial random variable.
01:30
Is given by this formula.
01:54
And so for part a we are asked for the probability that we must purchase exactly five boxes in order to obtain two prizes.
02:02
So this is the probability that x is equal to five.
02:07
And so using the probability of mess function, this is 4 choose 1 times 1 minus p, that's 0 .82 to the exponent 3 times p, that's 0 .18 to the exponent 2.
02:28
And this comes out to 0 .0715 approximately.
02:38
And then for b we are asked for the probability that we would have to purchase at least 10 boxes.
02:44
So this is the probability that x is at least 10...