Interesting Probability Questions1
Four Birthday Problems2
1. Assume that there are n people in the room. Ignoring leap years, what is the probability that no one
else in the room shares your birthday?
The general probability is: p = [364/365]n
2. Assume that there are 253 people in the room. Ignoring leap years, what is the probability that no
one else in the room shares your birthday? What is the probability that someone else in the room
shares your birthday?
Since: [364/365]253 = .499523 ˜ .5
Therefore: 1 - [364/365]253 = 1-.499523 ˜ .5
3. Assume that there are n people in the room. Ignoring leap years, what value of n (most closely)
makes the probability that someone else shares your birthday (1/n)?
Using: 1-[364/365]n = 1/n è n ˜ 19
4. Assume that there are n people in the room. Ignoring leap years, what value of n (most closely)
makes the probability that two people in the room share birthdays equal to .5?
Use: 1-[364/365][363/365][362/365]…[{365-(n-1)}/365]
since the nth person is the one we’re trying to find the same birthday for.
Let n=23: 1-[364/365][363/365][362/365]…[{365-22}/365] = .492703 ˜ .5
The Medical Diagnosis Problem3
5. Assume that a test to detect a disease whose prevalence is (1/1000) has a false positive rate of 5%
and a true positive rate of 100%. What is the probability that a person found to have a positive result
actually has the disease assuming that you know nothing about the person’s symptoms?
D = Has the disease
T = Test result is positive
Given:
P(D) = .001 so P(Not D) = .999
P(T | Not D) = .05
P(T | D) = 1.00
1All of these questions are common ones, but I took the suggestion to use them from: Richard G. Seymann. November
1991. “Comment”. The American Statistician. 45(4):287-8.
2 Due to: William Feller. 1957. An Introduction to Probability Theory and its Applications (Volume 1). New York: John
Wiley.
3 This question was presented to 60 students and staff at the Harvard Medical School. Over half of the participants
responded 95% and only 11 gave the correct response. Hmm… See: D. Kahneman, P. Slovik, and A. Tversky. 1982.
Judgement Under Uncertainty: Heuristics and Biases. Cambridge, UK: Cambridge University Press.
