Question

During Tim Hortons' annual "Roll Up the Rim to Win" promotion, customers who purchase a cup of coffee check under the rim of the cup to see if they have won a prize. It is known that 16.7% of all cups are winners. If you buy one cup of coffee every day for a week (7 days), what is the probability you win exactly one time?

Answer 1

Let's define the probability that any given cup of coffee is "winning" by

$$p_0 = 0.167. \\$$

Then the case that exactly one cup will "win" during seven purchases will be equal to intersection of 1 event of winning and 6 events of not winning:

$$p = p_0(1-p_0)^6 = 0.167~*~(1-0.167)^6 \approx 0.05579386915. \\$$

In percents,

$$p \approx 5.58 \%.$$