Question

The probability of buying a movie ticket with a popcorn coupon is 0.546 and without a popcorn coupon is 0.454. If you buy 27 movie tickets, we want to know the probability that exactly 15 of the tickets have popcorn coupons. (Consider tickets with popcorn coupons as successes in the binomial distribution.)

Answer 1

Answer:

$P(X=15)$

And using the probability mass function we got:

$P(X=15)=(27C15)(0.546)^{15} (1-0.546)^{27-15}=0.1523$

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we now that:

$X \sim Binom(n=27, p=0.546)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

And we want to find this probability:

$P(X=15)$

And using the probability mass function we got:

$P(X=15)=(27C15)(0.546)^{15} (1-0.546)^{27-15}=0.1523$