If I deal you one card from a standard deck of 52 and it is a face card, I will give you a $5.00 bill. There are 12 face cards. There is one catch: You have to pay me something to play this game. How much would that be if the game is to be fair?
Probability of winning = 12/52 = 3/13
Probability of losing = 40/52 = 10/13
Probability of losing = 3 1/3 times the probability of winning.
What would you consider fair?
X * 3 1/3 = 5
P(10 on 1st card and 3 on 2nd)
To determine the fair cost to play this game, we need to consider the probability of drawing a face card and the amount you would win.
First, let's calculate the probability of drawing a face card from a standard deck of 52 cards. There are 12 face cards (4 Jacks, 4 Queens, and 4 Kings) in a deck. So, the probability of drawing a face card is:
Probability of drawing a face card = Number of favorable outcomes / Total number of possible outcomes
Probability of drawing a face card = 12 / 52
Simplifying this, we get:
Probability of drawing a face card = 3 / 13
Now, let's determine the value of the $5.00 bill you would win. Since the game is supposedly fair, the expected value of your winning should be equal to the cost to play the game.
Expected value = (Probability of winning) * (Amount won) - (Probability of losing) * (Amount lost)
In this case, you have a probability of 3/13 to win $5.00, so the expected value would be:
Expected value = (3/13) * $5.00 - (10/13) * (cost to play)
Since the expected value should be zero (assuming a fair game), we can solve for the cost to play:
(3/13) * $5.00 - (10/13) * (cost to play) = 0
Simplifying this equation:
(3/13) * $5.00 = (10/13) * (cost to play)
Now, rearrange and solve for the cost to play:
(cost to play) = [(3/13) * $5.00] / (10/13)
Performing the calculations, we find:
(cost to play) = ($15.00 / 13)
So, the fair cost to play this game would be approximately $1.15.