AMC10 2005 B

AMC10 2005 B · Q12

AMC10 2005 B · Q12. It mainly tests Probability (basic), Primes & prime factorization.

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

掷 12 个公平的骰子。顶面数字的乘积是质数的概率是多少？

(A) $(\frac{1}{12})^{12}$ $(\frac{1}{12})^{12}$

(B) $(\frac{1}{6})^{12}$ $(\frac{1}{6})^{12}$

(D) $\frac{5}{2}(\frac{1}{6})^{11}$ $\frac{5}{2}(\frac{1}{6})^{11}$

(E) $(\frac{1}{6})^{10}$ $(\frac{1}{6})^{10}$

Answer

Correct choice: (E)

正确答案：（E）

Solution

Exactly one die must have a prime face on top, and the other eleven must have 1’s. The prime die can be any one of the twelve, and the prime can be 2, 3, or 5. Thus the probability of a prime face on any one die is 1/2, and the probability of a prime product is $12 \cdot (\frac{1}{2}) \cdot (\frac{1}{6})^{11} = (\frac{1}{6})^{10}$.

必须恰好有一个骰子顶面是质数，其他十一个必须是 1。质数骰子可以是 12 个中的任意一个，质数可以是 2, 3 或 5。因此单个骰子出现质数的概率是 $1/2$，质数乘积的概率是 $12 \cdot (\frac{1}{2}) \cdot (\frac{1}{6})^{11} = (\frac{1}{6})^{10}$。

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