Die Rolling Madness

Probability Level 2

If I roll a 6 sided die 4 times, what are the odds that the sum of all my rolls will be exactly 23?

83.33% 16.67% 0.31% 0.08%

Nik Olson by Nik Olson , 43, USA

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3 solutions

Gamal Sultan
Feb 10, 2015

The possible numbers for the sum to be 23 are 5,6,6,6

and this arrangement can be occur by 4 methods

(5,6,6,6), (6,5,6,6),(6,6,5,6),(6,6,6,5),

The all possible arrangements = 6^4 = 1296

The probability = 4/1296

The final answer = (4/1296)(100) = 0.30864 = 0.31%

11 Helpful 0 Interesting 0 Brilliant 0 Confused

I got this wrong the first time around because I calculated for 6 rolls (whoops), but here are two Python methods for finding the answer.

First, Monte Carlo for approximating: 

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from random import randint
def roll_23():
    total = 0
    for roll in xrange(4):
        total += randint(1,6)
    return total == 23
yes = 0
trials = 1000000
for trial in xrange(trials):
    if roll_23():
        yes += 1.0
print yes/trials

Secondly, for an exact answer, Cartesian product: 

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from itertools import product
yes = 0
trials = 0
sides = xrange(1, 7)
for rolls in product(sides,repeat=4):
    trials += 1
    if sum(rolls) == 23:
        yes += 1.0
print yes/trials

Brock Brown - 6 years, 3 months ago

I did't see we have the same solution.

Roman Frago - 6 years, 3 months ago
Roman Frago
Mar 12, 2015

4 6 4 = 0.30864 \frac {4}{6^4}=0.30864

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ruslan Abdulgani
Feb 8, 2015

The only possible arrangement for the sum of 23 are 6,6,6, 5. So there are (4!/3!) (1/6)4 100% = 0.31% chances.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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