There is a bag with balls numbered 1 to 9. Two players take turns at randomly taking a ball from the bag. If a player gets three balls that sum fifteen, he wins. Getting fifteen with more or less balls doesn't count.
What are the odds of the first player winning? Of a draw?
(In reply to
another approach -- solution by Charlie)
Here are my results, identical to what Charlie found.
Player One could win when drawing ball 5, 7, or 9
Player Two could win when drawing ball 6 or 8
5 34560 7 95904 9 81792 overall 212256
6 31968 8 72576 overall 104544
tie 46080
212256 104544
0.584920634920635 0.28809523809523807 0.12698412698412698
sum of all fractions 1.0
---------------------
from itertools import permutations
def winner(aList):
""" boolean: determine from a list or tuple of integers if 3 of them sum to 15 """
wins = [[1, 5, 9],
[1, 6, 8],
[2, 4, 9],
[2, 5, 8],
[2, 6, 7],
[3, 4, 8],
[3, 5, 7],
[4, 5, 6]]
if len(aList) < 3:
return False
for w in wins:
if w[0] in aList and w[1] in aList and w[2] in aList:
return True
return False
bs = [i for i in range(1,10)]
ways15 = []
waysNot = []
for perm in permutations(bs,3):
temp = sorted(list(perm))
if sum(perm) == 15:
if temp not in ways15:
ways15.append(temp)
else:
if temp not in waysNot:
waysNot.append(temp)
winOne = 0
winTwo = 0
winSeven = 0
winEight = 0
winNine = 0
draw = 0
for perm in permutations(bs,9):
first = perm[:3]
if winner(first):
winOne += 1
continue
second = perm[3:6]
if winner(second):
winTwo += 1
continue
seven = first + (perm[6],)
if winner(seven):
winSeven += 1
continue
eight = second + (perm[7],)
if winner(eight):
winEight += 1
continue
nine = seven + (perm[8],)
if winner(nine):
winNine += 1
continue
draw += 1
total = winOne + winTwo + winSeven + winEight + winNine + draw
OneWins = winOne + winSeven + winNine
TwoWins = winTwo + winEight
print('5',winOne, '7', winSeven, '9', winNine, 'overall', winOne+winSeven+winNine)
print('6',winTwo, '8', winEight, 'overall', winTwo+winEight)
print('tie',draw)
print(OneWins, TwoWins)
print(OneWins/total, TwoWins/total, draw/total)
print('sum of all fractions', OneWins/total+TwoWins/total+draw/total)
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Posted by Larry
on 2022-07-20 13:51:28 |
Confirmation of Charlie's results.
