In a casino game called high- low, there are three possible bets. Assume that $1
ID: 3150986 • Letter: I
Question
In a casino game called high- low, there are three possible bets. Assume that $1 is the amount of the bet you need to bet to play the game. A pair of fair six sided dice is rolled and their sum is calculated. If the bettor bets low he wins $1 if the sum of the dice is 2,3,4,5,6. If he bets high, he wins $1 if the sum of the dice is 8,9,10,11,12. If he bets on 7 he win $4 if sum of 7 is rolled. The bettor wins back his initial dollar he bet if he wins the bet, otherwise he loses it.
a) Find the expected value of the game to the bettor for all 3 bets. ( Hint: Find the probability weighted average of the gains and losses for each bet.). For example, for betting low, the gain would be $1 since he gets back his $1 bet. The losses are $1 for all bets.
b) Find the expected value of the game to the house if 200 bets of each are placed. ( Hint: the expected value to the house for each bet is the opposite of the expected value to the bettor. )
Explanation / Answer
Probabiity of occuring each pair of outcome (i,j) i=1(1)6, j=1(1)6 is 1/36 as there are 36 possible outcomes that can result and probability of each outcome is equal( as both the dice is fair).
Let S be the sum of the face value
S= 2 implies (1,1)
= 3 implies (1,2), (2,1)
= 4 implies (1,3), (3,1),(2,2)
= 5 implies (2,3),(1,4),(4,1),(3,2)
= 6 implies (1,5),(2,4),(3,3),(4,2),(5,1)
So total number of different cases = (1+2+3+4+5) = 15
So P(S=2,3,4,5,6) = 15/36 = 5/12
P(S=7) = P((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)) = 6/36 = 1/6
So P(S=8,9,10,11,12) = 1-(5/12+1/6) = 5/12
(a) for betting low
Gain $1 with prob 5/12 and loss $1 with prob (1-5/12) = 7/12
Expected gain = 1*5/12 - 1*7/12 = - $1/6
for betting high Expected gain = 1*5/12 - 1*7/12 = -$1/6
For betting 7 Expected gain = 4 *1/6 - 1*5/6 = -$1/6
So all the 3 bets expected gain is - $1/6 which is same.
(b) We see that for each type of bet expected value to the house is $1/6 whether it is low,high or 7
So expected value to the house if there is 200 bets of each type = 200*3*1/6 = $100
So the answer is $100 (answer)