Question

If four letters are placed into 4 addressed envelopes at random, the probability that exactly one letter will go wrong is
A. 0
B. $\dfrac{1}{2}$
C. $\dfrac{1}{3}$
D. $\dfrac{1}{4}$

Answer 1

Hint: To solve this question, we need to use the basic theory related to the probability. As we know the probability of an event will not be less than 0. And An event with a probability of zero means it is an impossible event. The probability of an event will not be more than 1. This is because 1 is certain that something will happen.

Complete step-by-step answer:
As given in the question,
four letters are placed into 4 addressed envelopes at random
therefore, Total arrangements = 4! = 24
If we put one letter in a wrong envelope, then we cannot manage to keep the other three correct. This is because if the other three are put in the correct envelopes, then the fourth one is left with the correct envelope too. Hence there is no such case when exactly one of them is correctly placed.
Hence P=0
To better convince ourselves that the above statement has some truth to it. We can test the result with a few examples.
An event with a probability of zero [P(E) = 0] will never occur (an impossible event).
Let’s say
1. The probability that tomorrow is Friday if today is Monday is 0.
2. The probability that you will be seventeen on your next birthday, if you were just born is 0.
In both cases, the probability of the event is Zero
Therefore, the probability that exactly one letter will go wrong is 0.
Thus, option (A) is the correct answer.

Note: Always remember that to calculate the probability of an event, we take the number of successes, divided by the number of possible outcomes as described below.
The probability of event A happening is:
P(A) = $\dfrac{{{\text{n}}\left( {\text{A}} \right)}}{{{\text{n}}\left( {\text{S}} \right)}}$