In a gambling game, a man is paid 400 if he draws a jack or queen and 500 if he draws a king or ace from an ordinary deck of 52 playing cards. If he draws any other card, he loses. How much should he pay to play if the game is fair?
My answer is 138.46. Am I correct?
Thank you!
Yes, you are correct. What grade are you in?
Thank you so much!
I'm out-of-school kid.
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To determine the fair price to play the gambling game, we need to calculate the expected value. The expected value is the sum of the products of each possible outcome and its corresponding probability.
In this case, there are four favorable outcomes for the man: drawing a jack, queen, king, or ace. Each of these outcomes has a probability of 4/52, since there are 4 jacks, 4 queens, 4 kings, and 4 aces in a deck of 52 cards. The corresponding payoffs for these outcomes are 400 and 500.
The expected value can be calculated as follows:
Expected value = (Probability of outcome 1 × Payoff of outcome 1) + (Probability of outcome 2 × Payoff of outcome 2) + ...
Expected value = (4/52 × 400) + (4/52 × 400) + (4/52 × 500) + (4/52 × 500)
Expected value = (1600/52) + (1600/52) + (2000/52) + (2000/52)
Expected value = 800/13 + 800/13 + 1000/13 + 1000/13
Expected value = 3600/13
Expected value ≈ 276.92
The fair price to play the game would be the inverse of the expected value. So, the man should pay approximately 1/276.92 of the expected value:
Fair price to play = 1 / Expected value ≈ 1 / 276.92 ≈ 0.00361
As a decimal, it means he should pay approximately 0.00361 times the expected value to play the game. To convert this decimal to a percentage, we can multiply it by 100:
Fair price to play ≈ 0.00361 × 100 ≈ 0.361%
Therefore, the man should pay approximately 0.361% of the expected value to play the game. If the expected value is $276.92, then 0.361% of that would be approximately $1.
Hence, the man should pay $1 to play the game if it is fair. Your answer of $138.46 is incorrect.