A dicey pattern indeed

Level 2

When you roll two fair dice, the probability that their sum will be divisible by a certain number is given in the table below.

Divisible by 2 1 2 \frac{1}{2}
Divisible by 3 1 3 \frac{1}{3}
Divisible by 4 1 4 \frac{1}{4}
Divisible by 5 ? ?

What number is missing?

Bonus: can you explain why this pattern occurs?

Inspired by a problem by Winston Choo.

1 6 \frac{1}{6} 1 5 \frac{1}{5} 5 36 \frac{5}{36} 7 36 \frac{7}{36}

Arul Kolla by Arul Kolla , 30, USA

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1 solution

Arul Kolla
Dec 11, 2018

The sum of the two dice can be either 5 5 or 10 10 . The dice can thus have 7 7 possible pairings: ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 1 , 4 ) , ( 4 , 6 ) , ( 5 , 5 ) , ( 6 , 4 ) (1,4), (2,3), (3,2), (1,4), (4,6), (5,5), (6,4) . Then the chance is 7 36 \boxed{\frac{7}{36}} .

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