Worksheet 11: Coincidences

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Sampling with and without replacement

1. Suppose a bag contains ten balls with labels 1 through 10. a. If you take three balls out of the bag and replace them each time, what is the size of the sample space? b. If you take three balls out of the bag do not replace them, what is the size of the sample space?

2. Suppose a bag contains ten balls with labels 1 through 10. You randomly take out three of the balls. a. If you replace the balls each time, what is the probability that you don’t draw a 1? b. If you do not replace the balls each time, what is the probability that you don’t draw a 1?

3. Suppose a bag contains ten balls with labels 1 through 10. You randomly take out ten of the balls. a. If you replace the balls each time, what is the probability that you don’t draw a 1? b. If you do not replace the balls each time, what is the probability that you don’t draw a 1?

Exclusive events

4. A and B are two events with \(\mathrm{Pr}[\text{A}]=0.7\) and \(\mathrm{Pr}[\text{B}] = 0.5\). Can the events A and B be exclusive?

5. In the Monty Hall Problem, should the contestant switch or stay with the original door?

6. Birthday problem a. What is the complement of “at least two people share a birthday”? b. What is the probability that two people have different birthdays? c. What is the probability that \(n\) people have different birthdays?

7. Streaks in basketball

   • Assume that the probability that a player makes a shot is \(p=0.47\).

   • Let \(n\) be the number basketball players.

   • What is the probability that at least one basketball player scores all \(k\) of their next shots?

8. Phone numbers

   a. What is the probability of getting a number like 111-1111 with all digits the same? b. What is the probability of getting a number like 358-6049 with all digits different?

9. What is the tennis simulation for Roger Federer missing?