Conditional Probability Questions

Question

Walt is selling candy outside the supermarket to raise money for new uniforms for the gymnastics team. The probability of a customer stopping to talk to Walt and buying some candy is 2/9 The probability of a customer just stopping to talk to Walt is 5/12. Fifty out of the 120 customers at the supermarket bought candy from Walt.

1.What is the probability of a customer buying candy from Walt given that they stopped to talk to him? Round to the nearest percent.

2.Are "a customer talking to Walt" and "a customer buying from Walt" independent or dependent events?

A survey has taken to determine the number of students that own a dog and a cat as a pet. When a student from the survey is chosen at random, the probability that the student owns both a dog and a cat is 1/12. When a student is chosen a random, the probability a dog is 1/4 and the probability that the student owns a cat is 1/3.

3.What is the probability that a student chosen at random who owns a dog also owns a cat? Round to the nearest percentage.

4.Are "owning a dog" and "owning a cat" independent or dependent events? 

Answer 1

Question 1

Event A: Stopping to talk to Walt

1. The probability of event A occurring is (5/12+2/9). Here the question intends somewhat to misdirect you. By adding these two numbers found in the statement, we find that this represents ALL customers who stopped to talk to Walt, whether they bought candy or not.
2. P(A) = 5/12+2/9 = 23/36 = 64%

Event B: Buying candy from Walt

1. The probability of event B occurring is 50/120 or 5/12, based on the number of customers that bought candy from Walt
2. P(B) = 50/120 = 5/12 = 42%

Event A & B: Stopping to talk to Walt AND buying candy from Walt

1. We are given that the probability of a customer stopping to talk to Walt and buying some candy is 2/9
2. P(A∩B) = 2/9 = 22%

Rewording the question in terms events A & B, we get: What is the probability of Event B given Event A P(B|A)?

Using the conditional probability formula, we have

P(B|A) = P(A∩B) / P(A)

Using our values and simplifying, we get: P(B|A)=(2/9) ⁄ (23/36) = 8/23 = 35%

Question 2

Events A and B are independent if P(B|A) = P(B) AND P(A|B) = P(A). Since we know that P(B|A) = 35% and P(B) = 42%, the first part of the statement above is incorrect, thus we know that that the two events are dependent. In other words, talking to Walt does affect whether or not the customer will buy candy.

Question 3

Let's apply the same principles here.

Event A: Owning a dog

1. P(A) = 1/4 = 25%

Event B: Owning a cat

1. P(B) = 1/3 = 33%

Event A & B: Owning a dog AND a cat

1. P(A∩B) = 1/12 = 8%

Rewording the question: What is the probability that given Event A, Event B also occurs P(B|A)?

P(B|A) = P(A∩B) / P(A) = (1/12) / (1/4) = 1/3 = 33%

Question 4

Since P(B|A) = P(B) = 33%, we now must check if P(A|B) = P(A)

P(A|B) = P(A∩B) / P(B) = (1/12) / (1/3) = 1/4 = 25% OR P(A|B) = P(A)

Thus, the conditions are satisfied and we have determined that the two events are independent. In other words, owning a dog has not effect on whether the student owns a cat.