Question

Your box of cereal may be a contest winner! It's rattling, which 100% of winning boxes do. Of course 1% of all boxes rattle and only one box in a million is a winner. What is the probability that your box is a winner? Please specify your answer as a probability (i.e. .05 rather than 5%).

Answer 1

Let's apply the Bayes theorem:

P(A|B) = P(B|A)*P(A)/P(B).

Here by A and B we denoted the following events:

event A - cereal box is winner
event B - cereal box rattles

One box in a million is a winner, so

P(A) = 10^(-6).

1% of all boxes rattle, so

P(B) = 10^(-2).

All of winning boxes rattles, so

P(B|A) = 1.

Therefore,

P(A|B) = P(B|A)*P(A)/P(B) = 1*10^(-6)/10^(-2) = 10^(-4).