Question

A new electronics store holds a contest to attract shoppers. Once an hour someone in the store is chosen at random to play the Music Game. Here's how it works: An ace and four other cards are shuffled and placed face down on a table. The customer gets to turn over cards one at a time, looking for the ace. The person wins $100 of store credit if the ace is the first card,$50 if it is the second card, and $20,$10, or $5 if it is the third, fourth, or last card chosen. What is the average dollar amount of store credit given away in the contest?

Answer 1

Let's calculate the average dollar amount of store credit given away in the Music Game.

The probabilities of finding the ace on the first, second, third, fourth, or last card are as follows:

- Probability of winning $100 (first card): \( \frac{1}{5} \)
- Probability of winning $50 (second card): \( \frac{4}{5} \times \frac{1}{4} = \frac{1}{5} \)
- Probability of winning $20 (third card): \( \frac{4}{5} \times \frac{3}{4} \times \frac{1}{3} = \frac{1}{5} \)
- Probability of winning $10 (fourth card): \( \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{1}{5} \)
- Probability of winning $5 (last card): \( \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} \times 1 = \frac{1}{5} \)

Now, let's calculate the average:

\[
\text{Average} = \left( \frac{1}{5} \times 100 \right) + \left( \frac{1}{5} \times 50 \right) + \left( \frac{1}{5} \times 20 \right) + \left( \frac{1}{5} \times 10 \right) + \left( \frac{1}{5} \times 5 \right)
\]

\[
\text{Average} = 20 + 10 + 4 + 2 + 1 = 37
\]

Therefore, the average dollar amount of store credit given away in the contest is $37.