Q: You are about to get on a plane to London, you want to know whether you have to bring an umbrella or not. You call three of your random friends and ask each one of them if it’s raining. The probability that your friend is telling the truth is 2/3 and the probability that they are playing a prank on you by lying is 1/3. If all 3 of them tell that it is raining, then what is the probability that it is actually raining in London.
You can tell that this question is related to Bayesian theory because of the last statement which essentially follows the structure, “What is the probability A is true given B is true?” Therefore we need to know the probability of it raining in London on a given day. Let’s assume it’s 25%.
P(A) = probability of it raining = 25%
P(B) = probability of all 3 friends say that it’s raining
P(A|B) probability that it’s raining given they’re telling that it is raining
P(B|A) probability that all 3 friends say that it’s raining given it’s raining = (2/3)³ = 8/27
Step 1: Solve for P(B)
P(A|B) = P(B|A) * P(A) / P(B), can be rewritten as
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(B) = (2/3)³ * 0.25 + (1/3)³ * 0.75 = 0.25*8/27 + 0.75*1/27
Step 2: Solve for P(A|B)
P(A|B) = 0.25 * (8/27) / ( 0.25*8/27 + 0.75*1/27)
P(A|B) = 8 / (8 + 3) = 8/11
Therefore, if all three friends say that it’s raining, then there’s an 8/11 chance that it’s actually raining.
