A game consists of cutting a shuffled deck of cards. If you cut a face card you win 10 cents. If you cut an ace you win 25 cents. If you cut anything else you lose a nickel. What is the expectation for this game?
To calculate the expectation for this game, we will need to determine the probability of each outcome and multiply it by the corresponding payout or loss. The expectation is then the sum of these values.
Let's start by calculating the probability of cutting a face card. There are 12 face cards in a standard deck (4 jacks + 4 queens + 4 kings) out of a total of 52 cards. Therefore, the probability of cutting a face card is 12/52, which simplifies to 3/13.
Next, let's calculate the probability of cutting an ace. There are 4 aces in a deck of cards, so the probability of cutting an ace is 4/52, which simplifies to 1/13.
Lastly, the probability of cutting anything else (i.e., a non-face card or non-ace) can be found by subtracting the sum of the probabilities of cutting a face card and an ace from 1. Since the probabilities of cutting face cards and aces are both 3/13 and their sum is 6/13, the probability of cutting anything else is 1 - 6/13, which simplifies to 7/13.
Now, let's calculate the expected value for each outcome:
- Cutting a face card: Probability (3/13) × Payout ($0.10) = $0.03
- Cutting an ace: Probability (1/13) × Payout ($0.25) = $0.0192 (rounded to four decimal places)
- Cutting anything else: Probability (7/13) × Loss (-$0.05) = -$0.027 (rounded to three decimal places)
To determine the expectation, we sum up the expected values for each outcome:
$0.03 + $0.0192 + (-$0.027) = $0.0222 (rounded to four decimal places)
Therefore, the expectation for this game is $0.0222, which means that on average, you can expect to earn $0.0222 per round of this game.