Question

Find the ace. 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 cards over one at a time, looking for the ace. The person wins \(\$ 100\) worth of free CDs or DVDs 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 fifth card chosen. What is the average dollar amount of music the store will give away?

Short answer

The average giveaway is $37 per game.

Step-by-step solution

List the values and probabilities

First, determine the dollar value the customer receives if the ace is found in each position and its respective probability:- First card: \(100 \( P(\text{Ace in 1st position}) = \frac{1}{5} \)- Second card: \)50 \( P(\text{Ace in 2nd position}) = \frac{1}{5} \)- Third card: \(20 \( P(\text{Ace in 3rd position}) = \frac{1}{5} \)- Fourth card: \)10 \( P(\text{Ace in 4th position}) = \frac{1}{5} \)- Fifth card: $5 \( P(\text{Ace in 5th position}) = \frac{1}{5} \)

Calculate expected payout

The expected value is calculated by multiplying each potential outcome by its probability and summing these products:\[E = (100 \times \frac{1}{5}) + (50 \times \frac{1}{5}) + (20 \times \frac{1}{5}) + (10 \times \frac{1}{5}) + (5 \times \frac{1}{5})\]Simplifying:\[E = 20 + 10 + 4 + 2 + 1 = 37\]

Final Answer

The expected giveaway amount per game is $37.

Key concepts

Probability

Probability is about measuring the likelihood of an event happening. When we talk about probability in the context of the "Find the Ace" game, we want to know the chances of picking the ace card in different positions.

• Each card has an equal right to be chosen as the ace.
• With five cards total, the probability of each card being the ace is \( \frac{1}{5} \).
• This is because only one of the five cards is the ace and they are shuffled randomly.

Probabilities are usually expressed as fractions, decimals, or percentages. In this example, \( \frac{1}{5} \) converts to a 0.20 probability or 20% chance. The uniform probability emphasizes fairness, as each position is equally possible due to the random shuffle of cards. This understanding allows us to predict average outcomes when the game is played repeatedly.

Random Selection

Random selection is a method used to ensure that each participant or element has an equal chance of being chosen. In the contest held by the electronics store, random selection is applied at two stages.

• First, when a customer is picked to play the game.
• Second, when the ace is shuffled among the cards.

Each customer present in the store has the same opportunity to participate in the game, making the process unbiased. Similarly, randomizing the cards ensures that each card position, from first to fifth, has an equal opportunity to be the ace.
This randomness protects fairness, as there is no pre-known or influenced order, maintaining integrity in the game. Because of this technique, the expected outcomes of the contest are reliable, averaging out over many repetitions to the expected value calculated.

Game Theory

Game theory analyzes competitive situations where the outcome depends on the actions of the participants involved. In the "Find the Ace" game, game theory can be used to understand the decision-making process involved.

• The store designs the game with payouts assigned to different positions, influencing players' perceptions of risk and reward.
• From a strategic perspective, the player might hope to increase their winnings by turning over cards thoughtfully.

Though the game seems straightforward, it still employs elements of game theory as it applies to probabilities. The store uses structured payouts to entice participants, optimizing allure without excessively risking profits.
The interplay of chance and strategy in this game reflects real-life applications of game theory, where decisions are made under uncertainty and with potential rewards or losses. However, it's important to understand that despite strategies, chance dominates due to the randomness of card positions.