AMC10 2015 B

AMC10 2015 B · Q11

AMC10 2015 B · Q11. It mainly tests Probability (basic), Primes & prime factorization.

Among the positive integers less than 100, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

在小于100的正整数中，其每个数字都是素数，从中随机选取一个。选中的数字是素数的概率是多少？

(A) $\frac{8}{99}$ $\frac{8}{99}$

(B) $\frac{2}{5}$ $\frac{2}{5}$

(D) $\frac{1}{2}$ $\frac{1}{2}$

(E) $\frac{9}{16}$ $\frac{9}{16}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

There are four one-digit primes (2, 3, 5, and 7), which can be used to form $4^2 = 16$ two-digit numbers with prime digits. Of these two-digit numbers, only 23, 37, 53, and 73 are prime. So there are $4 + 16 = 20$ numbers less than 100 whose digits are prime, and $4 + 4 = 8$ of them are prime. The probability is $\frac{8}{20} = \frac{2}{5}$.

有一位素数有4个（2、3、5和7），可以用它们组成$4^2 = 16$个两位素数数字的数。这些两位数中，只有23、37、53和73是素数。因此，小于100且数字都是素数的数有$4 + 16 = 20$个，其中$4 + 4 = 8$个是素数。概率是$\frac{8}{20} = \frac{2}{5}$。

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