1995 AMC 12 第 20 题

Question

若

a

、

b

、

c

是从集合

\{1,2,3,4,5\}

中有放回地随机选取的三个数（不一定互不相同），则

ab+c

为偶数的概率是

If

a,

b

and

c

are three (not necessarily different) numbers chosen randomly and with replacement from the set

\{1,2,3,4,5\},

the probability that

ab+c

is even is

\frac25

\frac{59}{125}

\frac12

\frac{64}{125}

\frac35

当

ab

与

c

奇偶性相同时，它们的和为偶数

The sum is even when

ab

and

c

have the same parity

只有当所选两个因数都是奇数时，乘积

ab

才是奇数

The product

ab

is odd only when both selected factors are odd

可选的奇数有

3

个，偶数有

2

个。乘积

ab

为奇数的概率是

\frac{9}{25}

，为偶数的概率是

\frac{16}{25}

。因此

\begin{aligned} P(ab+c\text{ 为偶数}) &=\frac9{25}\cdot\frac35\\ &\quad{}+\frac{16}{25}\cdot\frac25\\ &=\frac{59}{125} \end{aligned}\text{。}

所以正确答案是 B。

There are

3

odd and

2

even choices. The product

ab

is odd with probability

\frac{9}{25}

and even with probability

\frac{16}{25}.

Therefore

\begin{aligned} P(ab+c\text{ even}) &=\frac9{25}\cdot\frac35\\ &\quad{}+\frac{16}{25}\cdot\frac25\\ &=\frac{59}{125}. \end{aligned}

Thus the correct answer is B.

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