Worksheet 5: Coincidences#
Your name:
Your student ID number:
Robin’s RCT. You need to select a subset of \(n\) people to participate in an RCT; exactly one of these people is named Robin. You order them randomly, put the first \(m\) in a control group and the next \(m\) in the treatment group. The remaining people do not participate in the study.
a. What is the probability that Robin participates in the study?
b. What is the probability that Robin is in the control group?
c. What is the probability that Robin is in the treatment group?
d. What is the probability that Robin is \(k\)th in the random order?
e. Extra: If each person is instead selected to be in the control group independently with probability \(m/n\), and in case they didn’t make it, selected to be in the treatment group with probability \(m/n\), does Robin’s chance of being chosen for the study increase, decrease, or neither?
Headcount. You flip a fair coin \(n\) times.
a. What is the probability that it lands on heads every time?
b. What is the probability that it lands on heads exactly \(k\) times?
c. Extra: What is the probability that the number of heads is even? Find the simplest explanation that you can.
Birthday problem.
a. In a group of \(n\) uniformly random people, what is \(P(\ge 2\text{ people share a birthday})\)?
b. What is probability that someone shares your birthday?
Streakiness. Suppose every NBA player makes each shot independently with probability \(p\).
a. What is the probability that a player misses one of his next \(k\) attempted shots?
b. There are \(n\) players in the NBA. What is the probability that at least one of them makes all his next \(k\) attempted shots?
Phone numbers. Suppose phone numbers are chosen by choosing a random sequence of \(7\) digits in \(\{0,1,\ldots,9\}\).
a. Is it more likely that you are assigned the phone number 358-6049 or the phone number 111-1111?
a. What is the probability that all the digits are the same?
b. What is the probability that all the digits are different?
c. Extra: Is it more likely that all digits are the same, or that there are \(6\) distinct digits? (compare the probability of each event).