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: 
c. At least 5 believe that the minimum driving age should be 18.

Verified step by step guidance

1. Identify the type of probability distribution involved. 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 \(n = 15\), and the probability of success on each trial \(p = 0.60\) (the probability a woman believes the minimum driving age should be 18).
3. Express the event "at least 5 believe" mathematically as \(P(X \geq 5)\), where \(X\) is the binomial random variable representing the number of women who believe the minimum driving age should be 18.
4. Use the complement rule to simplify the calculation: \(P(X \geq 5) = 1 - P(X \leq 4)\). This means we will find the probability that 4 or fewer women believe the minimum driving age should be 18 and subtract it from 1.
5. Calculate \(P(X \leq 4)\) by summing the binomial probabilities for \(X = 0, 1, 2, 3,\) and \$4\( using the binomial probability formula: \[P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}\] where \)\binom{n}{k}\( is the binomial coefficient. Then subtract this sum from 1 to get \)P(X \geq 5)$.

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), with a probability of 0.60. The distribution helps calculate probabilities for different numbers of successes in the sample.

Probability of 'At Least' Events

Calculating the probability of 'at least' a certain number involves summing the probabilities of all outcomes from that number up to the maximum. For example, 'at least 5' means 5 or more women believe the minimum age should be 18, so we sum probabilities for 5, 6, ..., 15 successes in the binomial distribution.

Complement Rule

The complement rule states that the probability of an event occurring is 1 minus the probability that it does not occur. For 'at least 5' successes, it is often easier to calculate the complement probability of fewer than 5 successes and subtract from 1, simplifying the computation.