Bunuel wrote:
Patty and Selma play a gambling game in which Patty rolls a single fair six-sided die, numbered 1 through 6, and Selma rolls a single fair ten-sided die, numbered 1 through 10. If they tie, they reroll. Otherwise, the player with the higher number wins the loser’s bet. If Patty bets $5, how much should Selma bet to make the bet fair (so that each player will, on average, win the same amount of money)?
A. $8.33
B. $12
C. $13
D. $15
E. $18
Chance of Patty's winning:
Suppose Selma rolls 1, then patty can win in
5 different ways:by rolling one of 2,3,4,5, or 6
If Selma rolls 2, then patty can win in
4 different ways: by rolling one of 3,4,5, or 6
So we can conclude Patty can win in
5 + 4 + 3 + 2 + 1 = 15 different ways.
So probability of Patty's winning = \(\frac{15}{(6 * 10)}\) = \(\frac{15}{60}\)
[Here \(6 * 10 = 60\) is the number of total events that can occur by rolling a 6 sided dice and a 10 sided dice]
Chance of Selma's winning:
Suppose Patty rolls 1, then Selma can win in
9 different ways:by rolling one of 2,3,4,5,6,7,8,9,10
If Selma rolls 2, then Selma can win in
8 different ways:by rolling one of 3,4,5,6,7,8,9,10
Similarly we can conclude Selma can win in
9 + 8 + 7 + 6 + 5 + 4 = 39 different ways.
So probability of Selma's winning = \(\frac{39}{(6 * 10)}\) = \(\frac{39}{60}\).
As Patty bets $5 when Selma wins, Selma gets that $5. Let's assume Selma bets x amount of dollars. According to the question each player must win same amount in average. So we can say that,
\(x * \frac{15}{60} = 5 * \frac{39}{60}\)
\(x = 5 * \frac{39}{15} = 13\)
So
C is the answer