Probability of Yes and no answers

Question

Sara came across a new website that featured a "live" fortune-teller. She typed in three questions and found (shockingly) that the fortune-teller could only answer yes or no to her questions. Construct the sample space for all possible answers for her three questions. Assuming this spooky fortune-teller is actually just choosing the responses at random, find the probability that

1. All answers will be yes, or all answers will be no.
2. The answers will alternate (i.e., Yes-No-Yes or No-Yes-No).
3. Exactly three answers will be yes?

Write your answers in exact, simplified form.

Answer 1

You can use a tree diagram to help you with this problem. Start from a dot, and go to the 1st question. You can either get a "yes" or a "no" and since it's a random guess, the probability for either is 1/2. Then, you can go to the 2nd question and, no matter what happened on the 1st question, you can get a "yes" or a "no" and the probability for each is 1/2. Then, the same with the 3rd question:

So the sample space is:

YYY

YYN

YNY

YNN

NYY

NYN

NNY

NNN

1. The probability that all the questions will be answered "yes" is 1/8 and the probability that all the questions will be answered "no" is 1/8 so the probability that either one OR the other occurs is 1/8 + 1/8 = 2/8 = 1/4

1. The probability that the questions will be answered "YNY" is 1/8 and the probability that the questions will be answered "NYN" is 1/8 so the probability that either one OR the other occurs is 1/8 + 1/8 = 2/8 = 1/4

1. The probability that exactly 3 questions will be answered "yes" is 1/8

Answer 2