2021 AMC 12B Fall Problem 11

Question

Una rolls

6

standard

6

-sided dice simultaneously and calculates the product of the

6

numbers obtained. What is the probability that the product is divisible by

4?

\dfrac{3}{4}

\dfrac{57}{64}

\dfrac{59}{64}

\dfrac{187}{192}

\dfrac{63}{64}

Use complementary counting: the product has fewer than two factors of

2

Track whether each die contributes zero, one, or two factors of

2

The product fails to be divisible by

4

when it has at most one factor of

2.

Each die is odd with probability

\tfrac12,

contributes exactly one factor of

2

(a

2

or

6

) with probability

\tfrac13,

and two factors (a

4

) with probability

\tfrac16.

All six odd:

\left(\tfrac12\right)^6 = \tfrac{1}{64}.

Exactly one die a

2

or

6

and the rest odd:

6 \cdot \tfrac13 \cdot \left(\tfrac12\right)^5 = \tfrac{1}{16} = \tfrac{4}{64}.

The complement is

\tfrac{1}{64} + \tfrac{4}{64} = \tfrac{5}{64},

so the answer is

1 - \tfrac{5}{64} = \tfrac{59}{64}.

Thus, the correct answer is C.

Problem 11 in other years