Aaron offers Berenice to play a game of dice with him. Aaron explains the game to Berenice as follows:
"We each get one die, the highest one wins. If we tie, I win, but since you always lose when you roll a one, if you roll a one you can roll it again. If you get a one the second time you have to keep it."
What is each person's probability of winning? What are the probabilities of winning if Berenice can keep rolling until she gets something besides a one?
Original game:
Prob of # on roll:
A B
1 1/6 1/36 (i.e., prob of 1 both times for B)
2 1/6 1/6+1/36 (for 2 originally, or 1 then 2)
3 1/6 1/6+1/36 etc.
4 1/6 1/6+1/36
5 1/6 1/6+1/36
6 1/6 1/6+1/36
A's number B's prob of winning
1 35/36
2 28/36
3 21/36
4 14/36
5 7/36
6 0
Since all the probabilities of each number showing for A are equal, the average of the probabilities for B is the probability of her winning: (35+28+21+14+7+0)/(6*36) = 35/72 = 0.4861111111.... Of course A's probability of winning is 37/72.
Second version:
If B can roll until she gets something other than 1:
A B
1 1/6 0
2 1/6 1/5
3 1/6 1/5
4 1/6 1/5
5 1/6 1/5
6 1/6 1/5
A's number B's prob of winning
1 1
2 4/5
3 3/5
4 2/5
5 1/5
6 0
B's probability of winning is (5+4+3+2+1+0)/(6*5) = 5/10 = 1/2
Of course A's probability is also 1/2.
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Posted by Charlie
on 2022-11-14 10:40:18 |
solution
