The arena has a policy of selling as many as 16 tickets for a show for which there are only 15 seats available. The manager has noticed in the past that only 88% of people who purchase a ticket annually come for the show. Find the probability that if 16 tickets are sold, not enough seats will be available.
To find the probability that not enough seats will be available when 16 tickets are sold, we need to calculate the probability of more than 15 people showing up for the show.
First, let's calculate the probability that one individual who purchased a ticket will come to the show. Since only 88% of people come, the probability of a person showing up is 0.88.
Next, we can calculate the probability that more than 15 people will show up out of the 16 tickets sold. To do this, we'll use the binomial probability formula:
P(X > 15) = 1 - P(X ≤ 15)
where X is a binomial random variable representing the number of people who show up out of the 16 tickets sold.
Now, let's substitute the values into the formula:
P(X > 15) = 1 - P(X ≤ 15)
= 1 - Σ [P(X = x)] for x = 0 to 15
To calculate P(X = x), we use the binomial probability formula:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
where n is the total number of trials, p is the probability of success in each trial, and nCx is the combination formula.
In our case, n = 16, p = 0.88, and we need to calculate the probability for x = 0 to 15.
Now, let's calculate the probability for each value of x and sum them up:
P(X > 15) = 1 - Σ [P(X = x)] for x = 0 to 15
After calculating the individual probabilities for each x, we can substitute them into the formula and find the final result.