Four Dice

Probability Level 3

Four dice are thrown. What is the probability that the product of the numbers is equal to 36? If the answer can be expressed as a b \frac ab where a , b a, b are relatively coprime positive integers, what is the value of a + b a+b ?


The answer is 28.

Paul Fournier by Paul Fournier , 70, Canada

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2 solutions

Randy Yap
Jun 24, 2015

There are 6^4 combinations or 1296 to throw 4 dices. The number of ways to get a product of 36 from those numbers are as follows: We can throw a combination of {1,1,6,6}, {2,2,3,3},{1,3,3,4} , {1,2,3,6} .

For {1,1,6,6} there are 4!/(2!2!)=6 ways

For {2,2,3,3,} there are 4!/(2!2!) = 6 ways

For {1,3,3,4} there are 4!/2!=12 ways

For {1,2,3,6} there are 4!=24 ways.

Hence there are a total of 48 ways to throw the dices where the product of these numbers equate to 36. The probability is then 36/1296 or 1/27

7 Helpful 0 Interesting 0 Brilliant 0 Confused
Brock Brown
Jul 13, 2015

Python 3.3: 

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from itertools import product as combos
from fractions import Fraction as frac
from functools import reduce
goals = 0
total = 0
for throw in combos(range(1,7), repeat = 4):
    product = reduce(lambda a,b: a*b, throw, 1)
    if product == 36:
        goals += 1
    total += 1
answer = frac(goals, total)
print ("Answer:", answer.numerator + answer.denominator)

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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