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1. 甲乙丙丁四人互相给其他的三人之一写信，选择对象的方式是等可能的. 问存在两个人收到对方的信的概率（） A.

\frac{2}{3}

B. $\fr...

Mar 9, 2024

1. 甲乙丙丁四人互相给其他的三人之一写信，选择对象的方式是等可能的. 问存在两个人收到对方的信的概率（） A.

\frac{2}{3}

\frac{5}{9}

\frac{17}{27}

\frac{55}{81}

Solution by Steps

step 2

To find the number of ways two specific people can send letters to each other, we fix their choices and consider the choices of the other two people. Each of the other two people has 2 choices, so there are

2^2

ways for this scenario

step 3

Since there are 4 people, there are

\binom{4}{2}

or 6 possible pairs of people who could potentially send letters to each other

step 4

Multiply the number of ways for one specific pair to send letters to each other by the number of possible pairs to get the total number of favorable outcomes:

6 \times 2^2

step 5

Divide the number of favorable outcomes by the total number of ways to find the probability:

\frac{6 \times 2^2}{3^4}

step 6

Simplify the fraction to get the probability:

\frac{6 \times 4}{81} = \frac{24}{81} = \frac{8}{27}

. Since this option is not available in the multiple choices, we need to consider that the probability calculated is for exactly one pair exchanging letters, not at least one pair

step 7

To find the probability of at least one pair exchanging letters, we need to subtract the probability of no pairs exchanging letters from 1

step 8

The probability of no pairs exchanging letters is the complement of at least one pair exchanging letters. This is the same as each person sending a letter to someone who is not the person they received a letter from. There are 9 ways for this to happen (derangements of 4 items)

step 9

Calculate the probability of no pairs exchanging letters:

\frac{9}{3^4}

step 10

Subtract this probability from 1 to get the probability of at least one pair exchanging letters:

1 - \frac{9}{81} = 1 - \frac{1}{9} = \frac{8}{9}

step 11

The probability of at least one pair exchanging letters is

\frac{8}{9}

, which is not in the simplest form. We need to find the probability of exactly one pair exchanging letters

step 12

To find the probability of exactly one pair exchanging letters, we subtract the probability of two pairs exchanging letters from the probability of at least one pair exchanging letters. There is only 1 way for two pairs to exchange letters (each pair sends a letter to each other)

step 13

Calculate the probability of two pairs exchanging letters:

\frac{1}{3^4}

step 14

Subtract this probability from the probability of at least one pair exchanging letters:

\frac{8}{9} - \frac{1}{81} = \frac{72}{81} - \frac{1}{81} = \frac{71}{81}

step 15

The probability of exactly one pair exchanging letters is

\frac{71}{81}

, which simplifies to

\frac{71}{81}

. This is not one of the options, indicating a mistake in the calculation. We need to reconsider the approach to find the correct probability

Key Concept

Probability of at least one pair exchanging letters

Explanation

The probability of at least one pair exchanging letters is calculated by considering the total number of ways letters can be exchanged and subtracting the number of ways no pairs exchange letters.