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I wrote a letter to my friends X and gave it to my son to post it. The probability that my son will forget to post the letter is $\dfrac{1}{{10}}$ and the letter will be lost in the post in $\dfrac{1}{{100}}$. If my friend X did not receive the letter, then the probability that
(A) my son forgot to post the letter is $\dfrac{{891}}{{1000}}$
(B) the letter was lost in the post $\dfrac{{891}}{{991}}$
(C) my son forgot to post the letter is $\dfrac{{100}}{{991}}$
(D) none of these

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Last updated date: 22nd Mar 2024
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MVSAT 2024
Answer
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Hint:
For finding the probability that the son forgot to post the letter while we already know that letter does not receive by person X, we need to use the conditional probability. We need to separately find the probability that the son forgot to post the letter and divide it with the probability that the letter did not get received by person X. For calculating the probability of letter not being received by person X we can add up the probability of son forgetting to post the letter and the probability of son posting the letter and the letter getting lost in the post.

Complete step by step solution:
Here in this problem, we are given a situation of conditional probability. In this situation, a letter is written to a person X and another person is asked to post that letter. The probability that the person will forget to post the letter is given as $\dfrac{1}{{10}}$ and the probability that the letter gets lost in the post is given as $\dfrac{1}{{100}}$. Now if the person X didn’t get the letter, then we need to find the probability according to the options.
So, we can understand it as if the person posts the letter then after that the probability of losing the letter in the post comes into play. If the person forgets to post the letter then the letter won’t get posted.
$ \Rightarrow $ The probability that letter does not reach the person X $ = $ Probability that the son forgets to post$ + $ Probability that son posts the letter but it gets lost in the post
As we know that the complement of the probability of an event can be calculated by subtracting the probability of that event from one. So, the probability of the son posting the letter will be the probability of son forgetting to post subtracted from one, i.e. $1 - \dfrac{1}{{10}} = \dfrac{9}{{10}}$
$ \Rightarrow $ Probability of letter not reaching the person X$ = \dfrac{1}{{10}} + \left( {1 - \dfrac{1}{{10}}} \right) \times \dfrac{1}{{100}} = \dfrac{1}{{10}} + \dfrac{9}{{1000}} = \dfrac{{109}}{{1000}}$
Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.
$ \Rightarrow $ Probability of son forgetting to post letter and letter doesn’t get delivered$ = \dfrac{{{\text{Probability of son forgetting to post}}}}{{{\text{Probability of person X not getting letter}}}} = \dfrac{{\dfrac{1}{{10}}}}{{\dfrac{{109}}{{1000}}}} = \dfrac{1}{{10}} \times \dfrac{{1000}}{{109}} = \dfrac{{100}}{{109}}$
Now for checking the option, let’s solve for option (B)
We need to calculate the probability of a letter being lost in the post if the letter does not get delivered.
So the probability of letter getting lost in the post will be the product of the probability of son posting the letter and the probability of letter lost in the post
$ \Rightarrow $ Probability letter getting lost in the post $ = \dfrac{9}{{10}} \times \dfrac{1}{{100}} = \dfrac{9}{{1000}}$
So, the probability of letter getting lost in the post if the person X didn’t receive a letter $ = \dfrac{{\dfrac{9}{{1000}}}}{{\dfrac{{109}}{{1000}}}} = \dfrac{9}{{109}}$
Therefore, we got the required probability but the value of probability in the option doesn’t match the value we found.

Hence, the option (D) is the correct answer.

Note:
Remember that the use of the concept of conditional probability was a crucial part of the solution. Conditional probability is the probability of an event occurring given that another event has already occurred. The concept is one of the quintessential concepts in probability theory. Note that conditional probability does not state that there is always a causal relationship between the two events, as well as it does not indicate that both events occur simultaneously.