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# Patty and Selma play a gambling game in which Patty rolls a single fai

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Math Expert
Joined: 02 Sep 2009
Posts: 64144
Patty and Selma play a gambling game in which Patty rolls a single fai  [#permalink]

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18 Nov 2018, 22:22
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24
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Difficulty:

95% (hard)

Question Stats:

24% (02:18) correct 76% (02:05) wrong based on 85 sessions

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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

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Re: Patty and Selma play a gambling game in which Patty rolls a single fai  [#permalink]

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19 Nov 2018, 23:12
5
6
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

Total cases = 6 * 10 = 60

Cases in which Patty wins = 1 + 2 + 3 + 4 + 5 = 15
Patty 2, Selma 1
Patty 3, Selma 1/2
...
Patty 6, Selma 1/2/3/4/5

Case in which no one wins = 6 (Both throw the same number 1/2/3/4/5/6)

Cases in which Selma wins = 60 - 15 - 6 = 39

Since Patty wins in 15 cases and Selma wins in 39, they should bet money in the same ratio so that they win an equal amount on average.
If Patty bets $5, Selma should bet$13.
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Patty and Selma play a gambling game in which Patty rolls a single fai  [#permalink]

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Updated on: 20 Nov 2018, 08:58
5
5
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 in5 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,
$$x * \frac{15}{60} = 5 * \frac{39}{60}$$
$$x = 5 * \frac{39}{15} = 13$$

Originally posted by sarthoks on 19 Nov 2018, 21:57.
Last edited by sarthoks on 20 Nov 2018, 08:58, edited 1 time in total.
##### General Discussion
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Re: Patty and Selma play a gambling game in which Patty rolls a single fai  [#permalink]

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19 Nov 2018, 02:16
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

Good question took me over 2 mins to solve i hope that answer is correct...

P of Patty winning is 1/6 a
P of Selma winning is 1/6 *4/10 ; 4/60 : as she has 10 sided die so (7,8,9,10) are all her winning no.

Avg of winning of both : 4/60+1/6= 14/60= 7/30
and avg of loosing the game of both =23/30
so the amount which needs to be added so as to have a fair bet = (23/30)*5= 3.33 additional amount
so total amount which Selma would have to add is 5+3.33 $= 8.33$ option A IMO...

GMATinsight : sir please check if the working & answer is okay or any other alternate solution is available for this question...
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Re: Patty and Selma play a gambling game in which Patty rolls a single fai  [#permalink]

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20 Nov 2018, 01:03
1
Three possibilities:
#1: P wins
#2: Draw
#3: S wins

Case 1 Probability: P wins
P gets 2 & S gets 1= 1/6*1/10= 1/60
P gets 3 & S gets 1 or 2= 2/60
P gets 4= 3/60
P gets 5= 4/60
P gets 6= 5/60
Probability (P wins)= 15/60

Case 2 Probability: A Draw
P gets 1 & S gets 1= 1/6*1/10= 1/60
Total 6 such cases, each with Probability= 1/60
Probability (Draw)= 6/60

Case 3 Probability: S wins
Probability (S wins)= 1-(15/60 + 6/60) = 39/60

Now, out of 60, if S wins 39 games @ $5 per game, then Total Winning Amount = 39*5=$195

To win this amount of $195 in 15 games, P must win @ 195/15 =$13 per game.

So, in order to have an equal opportunity to win same money, S must bet $13 per game. Hence, Ans C Posted from my mobile device Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 10566 Location: United States (CA) Re: Patty and Selma play a gambling game in which Patty rolls a single fai [#permalink] ### Show Tags 15 Mar 2020, 03:53 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 Since one die is 6-sided and the other die is 10-sided, there are 6 x 10 = 60 possible outcomes for the roll of the two dice. Let’s calculate the number of wins, ties, and losses for the two players. (For example, if Patty rolls a 1, she cannot win. There is one way for a tie, if Selma also rolls a 1. Otherwise, Selma will win with any roll from 2 to 10, which is 9 times.) Patty rolls a 1. Patty wins: 0. Ties: 1 Selma wins: 9 Patty rolls a 2. Patty wins 1. Ties: 1 Selma wins: 8 Patty rolls a 3. Patty wins 2. Ties: 1 Selma wins: 7 Patty rolls a 4. Patty wins 3. Ties: 1 Selma wins: 6 Patty rolls a 5. Patty wins 4. Ties: 1 Selma wins: 5 Patty rolls a 6. Patty wins 5. Ties: 1 Selma wins 4 Out of 60 rolls, Patty wins 15, there are 6 ties, and Selma wins 39. If we want a fair game for both players, we can make the bets proportional to their respective probabilities of winning. Since Patty’s bet is$5, and the ratio of wins (Selma to Patty) is 39 : 15, which is equivalent to 13 : 5, we have:

13/5 = x/5

13 = x

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Re: Patty and Selma play a gambling game in which Patty rolls a single fai   [#permalink] 15 Mar 2020, 03:53