Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that: 
a. Exactly 10 believe that the minimum driving age should be 18.

Verified step by step guidance

1. Identify the type of probability distribution to use. Since we are dealing with a fixed number of independent trials (15 women), each with two possible outcomes (believe or not believe the minimum driving age should be 18), this is a binomial distribution problem.
2. Define the parameters of the binomial distribution: the number of trials \(n = 15\), the probability of success on each trial \(p = 0.60\) (since 60% believe the minimum driving age should be 18), and the number of successes \(k = 10\) (exactly 10 women).
3. Write down the binomial probability formula to find the probability of exactly \(k\) successes in \(n\) trials: \[P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}\] where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to choose \(k\) successes from \(n\) trials.
4. Calculate the binomial coefficient \(\binom{15}{10}\), which counts the number of ways to select 10 women who believe the minimum driving age should be 18 out of 15 women.
5. Substitute the values into the formula: compute \(p^{10}\), \((1-p)^{5}\), and multiply by the binomial coefficient to get the probability of exactly 10 women believing the minimum driving age should be 18.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, each woman either believes the minimum driving age should be 18 (success) or not (failure), making this a binomial scenario with n = 15 and p = 0.60.

Probability Mass Function (PMF) of Binomial Distribution

The PMF gives the probability of exactly k successes in n trials and is calculated using combinations and the success probability. For this problem, it helps find the probability that exactly 10 out of 15 women support the driving age of 18.

Combinations (n choose k)

Combinations count the number of ways to choose k successes from n trials without regard to order. This is essential in the binomial formula to determine how many different groups of 10 supporters can be formed from 15 women.