Question

Billy’s mom baked thirty chocolate chip cookies this morning. After letting them cool, she added them to the cookie jar on the kitchen table, which already contained six peanut butter cookies. Ten minutes later, Billy sneaks into the kitchen, climbs up onto a chair, and pulls three cookies out of the cookie jar. What is the probability that Billy gets exactly one peanut butter cookie?

Answer 1

To find the probability of Billy getting exactly one peanut butter cookie, we calculate the outcomes where one peanut butter cookie is drawn among three taken from the jar. We determine the possible arrangements and then calculate the probability for each scenario, multiplying by three for the total probability.

Step-by-step explanation:

Billy’s mom baked thirty chocolate chip cookies and added them to a jar with six peanut butter cookies. After Billy sneaks three cookies, we're asked to calculate the probability that he gets exactly one peanut butter cookie. First, we need to determine the total number of cookies, which is 36 (30 chocolate chip + 6 peanut butter).

There are several combinations for Billy to get exactly one peanut butter cookie when he takes three cookies out:

1. He gets a peanut butter cookie first, then two chocolate chip cookies.
2. He gets a chocolate chip cookie, then a peanut butter cookie, and another chocolate chip cookie.
3. He gets two chocolate chip cookies and then a peanut butter cookie.

The probability of Billy taking one peanut butter cookie is the sum of the probabilities for each scenario. The probability for scenario 1 would be × (÷ ÷ ). Scenario 2 and 3 would have a similar approach but different order, resulting in the same product. Thus, the final probability is given by 3 times the probability calculated for one scenario.

Step-by-step calculation for scenario 1:

• P(peanut butter first) = 6/36
• P(chocolate chip second) = 30/35 (since there's now one less peanut butter cookie)
• P(chocolate chip third) = 29/34 (since there's now one less chocolate chip cookie)

The final probability for this exact scenario is (6/36) × (30/35) × (29/34). Multiply this by 3 to account for each scenario. The complete calculation gives the probability of Billy getting exactly one peanut butter cookie.