2021 AMC 12B Spring 第 19 题

Question

掷两个公平骰子，每个骰子至少有

6

个面。每个骰子的各面分别标着从

1

到该骰子面数的不同整数。掷出和为

7

的概率是掷出和为

10

的概率的

\dfrac34

，且掷出和为

12

的概率是

\dfrac{1}{12}

。两个骰子的面数总和最小可能是多少？

Two fair dice, each with at least

6

faces are rolled. On each face of each die is printed a distinct integer from

1

to the number of faces on that die, inclusive. The probability of rolling a sum of

7

is

\dfrac34

of the probability of rolling a sum of

10,

and the probability of rolling a sum of

12

is

\dfrac{1}{12}.

What is the least possible number of faces on the two dice combined?

16

17

18

19

20

答案：B

知识点：骰子（概率）最优化

难度评级：2120

小提示：

因为每个骰子至少有

6

个面，和为

7

总有

6

种方式，所以和为

10

有

8

种方式

With at least

6

faces each, a sum of

7

always has

6

ways, so a sum of

10

has

8

ways

大提示：

设两个骰子分别有

a\le b

个面；数和为

10

与

12

的结果数，并使用

\text{方法数}(12)=\tfrac{ab}{12}

Let the dice have

a\le b

faces; count outcomes for sums

10

and

12

and use

\text{ways}(12)=\tfrac{ab}{12}

解答：

设两个骰子的面数为

a\le b

。因为两个骰子都至少有

6

个面，和为

7

恰有

6

种方式，所以和为

10

有

6\div\tfrac34=8

种方式。

掷出

10

的方式数为

\min(a,9)

-\max(1,10-b)+1=8

。和为

12

的概率为

\tfrac{1}{12}

，所以它有

\tfrac{ab}{12}

种方式。

和为

10

有

8

种结果，要求

a\ge8

且

b\ge9

，所以必有

a+b\ge17

。当

(a,b)=(8,9)

时，和为

10

恰有

8

种结果，而和为

12

恰有第一枚骰子掷出

3,4,\ldots,8

时的

6

种结果。因为

\frac{6}{8\cdot9}=\frac{1}{12}

，两个条件都成立，从而下界

17

可以达到。

所以正确答案是 B。

Let the dice have

a\le b

faces. Since both have at least

6

faces, a sum of

7

occurs in exactly

6

ways, so a sum of

10

occurs in

6\div\tfrac34=8

ways.

The number of ways to roll

10

is

\min(a,9)

-\max(1,10-b)+1=8.

A sum of

12

has probability

\tfrac{1}{12},

so it occurs in

\tfrac{ab}{12}

ways.

Having

8

outcomes for a sum of

10

requires

a\ge8

and

b\ge9,

so necessarily

a+b\ge17.

For

(a,b)=(8,9),

a sum of

10

has

8

outcomes, while a sum of

12

has the

6

outcomes with the first die showing

3,4,\ldots,8.

Since

\frac{6}{8\cdot9}=\frac{1}{12},

both conditions hold and the lower bound

17

is attained.

Thus, the correct answer is B.

2021 AMC 12B Spring 第 19 题

答案：B

难度评级：2120

小提示：

大提示：

解答：

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