Probability of Paying in Cash at David's Gasoline Station

E

P(20) is almost zero. Probabilities of higher than 20 are also close to zero. So the probability of not more than 20 is almost 1.

To solve this problem, we need to use the binomial probability formula. The formula for the probability of obtaining exactly k successes in n independent Bernoulli trials is:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Where:

• n is the number of trials (number of customers)
• k is the number of successes (number of customers paying in cash)
• p is the probability of success on each trial (probability of a customer paying in cash)
• (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials and is calculated as (n! / (k! * (n - k)!))

In this case, we know that 40% of all customers pay in cash, so the probability of a customer paying in cash (p) is 0.4. We also know that there were twenty-five customers (n) in the one-hour period.

Now, let's calculate the probability that no more than twenty customers pay in cash:

P(X ≤ 20) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 20)

To calculate this, we can use the binomial probability formula for each value of k from 0 to 20 and sum them up.

P(X = 0) = (25 choose 0) * (0.4)^0 * (1 - 0.4)^(25 - 0) = (1) * (1) * (0.6)^25

P(X = 1) = (25 choose 1) * (0.4)^1 * (1 - 0.4)^(25 - 1) = (25) * (0.4) * (0.6)^24

We repeat this calculation for each value of k from 2 to 20.

After calculating the probabilities for each value of k, we sum them up to get the final result:

P(X ≤ 20) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 20)

Comparing this probability to the answer choices, we can determine the correct answer.

Note: Since this is an exam question, it's important to show your work and calculations explicitly to receive full credit. The detailed explanation provided here should help you understand the approach to solving the problem.