Question

You are playing a card game. If you pull an Ace you win $26, if you pull the King of hearts you win $204, if you pull a jack or a queen you win $13, all other pulls you lose. What price should you be charged to make this a fair game?
$5
$7
$6
$8​

Answer 1

Using the expected value, it is found that a price of $6 should be charged to make it a fair game.

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• The game is fair if the expected value is 0.
• The expected value is given by each outcome multiplied by it's probability.
• A probability is the number of desired outcomes divided by the number of total outcomes.

• In a standard deck, there are 52 cards.
• The cost of x.

• There are 4 Aces in a standard deck, thus $\frac{4}{52}$ probability of earning 26 - x.
• There is 1 King of Hearts in a standard deck, thus $\frac{1}{52}$ probability of earning 204 - x.
• There are 4 Jacks and 4 Queens in a standard deck, thus $\frac{8}{52}$ probability of winning 13 - x.
• The other 39 cards you lose, thus $\frac{39}{52}$ probability of losing x.

Considering we want the expected value to be 0.

$\frac{4}{52}(26 - x) + \frac{1}{52}(204 - x) + \frac{8}{52}(13 - x) - \frac{39}{52}x = 0$

$\frac{4(26 - x) + 204 - x + 8(13 - x) - 39x}{52} = 0$

$104 - 4x + 204 - x + 104 - 8x - 39x = 0$

$52x = 312$

$x = \frac{312}{52}$

$x = 6$

$6 should be charged.

A similar problem is given at brainly.com/question/24905256