Suppose you had written 50 letters and addressed 50 envelopes and then inserted the letters randomly.

What is the probability that exactly one letter goes to the wrong person
(ii) What is the probability that each letter goes to the correct person? Justify your decision.

To solve this problem, we will use the concept of permutations and combinations.

(i) To find the probability that exactly one letter goes to the wrong person, we need to determine the total number of ways that exactly one letter can go to the wrong person, and then divide it by the total number of possible arrangements of the letters.

1. First, let's consider the number of ways that exactly one letter can go to the wrong person. There are a total of 50 letters, and for each one of them, there are 49 other envelopes to choose from. Therefore, there are 50 different ways to choose the letter that gets misaddressed.

2. Next, let's consider the number of ways to arrange the remaining 49 letters and envelopes correctly. We can think of this as arranging 49 objects in their corresponding positions. The number of ways to do this is 49!.

3. Finally, the total number of possible arrangements of the 50 letters and envelopes is 50!.

Therefore, the probability that exactly one letter goes to the wrong person is given by:

P(exactly one letter wrong) = (50 * 49!) / 50!

Simplifying this expression, we find:

P(exactly one letter wrong) = 1/50

So, the probability that exactly one letter goes to the wrong person is 1/50.

(ii) To find the probability that each letter goes to the correct person, we can use the concept of derangements. A derangement is an arrangement where none of the elements appear in their original positions.

The number of derangements of a set of n elements denoted D(n) can be calculated using the formula:

D(n) = n! * (1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)

For n = 50, the number of derangements is:

D(50) = 50! * (1 - 1/1! + 1/2! - ... + (-1)^50/50!)

Therefore, the probability that each letter goes to the correct person is given by:

P(each letter correct) = D(50) / 50!

To determine this probability accurately, we can use a calculator or a computer program to calculate D(50) and 50!. However, it is a known fact that as n approaches infinity, D(n) approaches 1/e (approximately 0.3678794), where e is Euler's number (approximately 2.71828).

Based on this property, since n = 50 is a large number, we can approximate the probability as follows:

P(each letter correct) ≈ 1/e ≈ 0.3678794

Therefore, the probability that each letter goes to the correct person is approximately 0.3679 or 36.79%.

In summary:
(i) The probability that exactly one letter goes to the wrong person is 1/50.
(ii) The probability that each letter goes to the correct person is approximately 0.3678794 or 36.79%.