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 in advance!
To determine the fair price to play the game, we need to calculate the expected value. The expected value, in this case, is the sum of the probabilities of each outcome multiplied by the corresponding payoff.
There are four favorable cards (jack, queen, king, ace) out of 52 cards in a deck.
The probability of drawing a jack or queen is 4/52, as each suit contains 2 jacks and 2 queens.
The probability of drawing a king or ace is also 4/52.
The probability of drawing any other card is 44/52, as there are 48 cards that are not jacks, queens, kings, or aces.
Now, let's calculate the expected value:
Expected Value = (Probability of Drawing Jack or Queen) * (Payoff for Jack or Queen)
+ (Probability of Drawing King or Ace) * (Payoff for King or Ace)
+ (Probability of Drawing Other Card) * (Payoff for Other Card)
Expected Value = (4/52) * 400 + (4/52) * 500 + (44/52) * (-1)
Expected Value = (16/52) * 400 + (16/52) * 500 - (44/52)
Expected Value = 12.308 + 15.385 - 0.846
Expected Value = 26.847
Since the expected value is positive (greater than zero), this means the game is favorable for the player. Therefore, the fair price to play is equal to the expected value, which is $26.847.
Hence, the answer is not 138.46. The correct answer is $26.847.