game in a casino consists of drawing one card from a standard deck of cards. If the player Draws an ace, they win €20Draws a face card(Jack, Queen,King) they win €10Draws a 10, they win €5Draws a 2, they lose €5Draws anything else(3 - 9), they win nothingCalculate the expected value of the game (Give your answer correct to 2 decimal places)
Question
game in a casino consists of drawing one card from a standard deck of cards. If the player Draws an ace, they win €20Draws a face card(Jack, Queen,King) they win €10Draws a 10, they win €5Draws a 2, they lose €5Draws anything else(3 - 9), they win nothingCalculate the expected value of the game (Give your answer correct to 2 decimal places)
Solution
To calculate the expected value of the game, we need to multiply the value of each outcome by the probability of that outcome, and then sum these values.
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Drawing an Ace: There are 4 aces in a deck of 52 cards. So, the probability of drawing an ace is 4/52 or 1/13. The payout is €20. So, the expected value from drawing an ace is (1/13)*€20 = €1.54
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Drawing a face card (Jack, Queen, King): There are 12 face cards in a deck. So, the probability of drawing a face card is 12/52 or 3/13. The payout is €10. So, the expected value from drawing a face card is (3/13)*€10 = €2.31
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Drawing a 10: There are 4 tens in a deck. So, the probability of drawing a 10 is 4/52 or 1/13. The payout is €5. So, the expected value from drawing a 10 is (1/13)*€5 = €0.38
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Drawing a 2: There are 4 twos in a deck. So, the probability of drawing a 2 is 4/52 or 1/13. The player loses €5 in this case. So, the expected value from drawing a 2 is (1/13)*-€5 = -€0.38
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Drawing anything else (3-9): There are 28 such cards in a deck. So, the probability of drawing a 3-9 is 28/52 or 7/13. The payout is €0. So, the expected value from drawing a 3-9 is (7/13)*€0 = €0
Adding these expected values together, the total expected value of the game is €1.54 + €2.31 + €0.38 - €0.38 + €0 = €3.85
So, the expected value of the game is €3.85.
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