Discussion
Sanjay writes a letter to his friend, Kesari. It is known that one out of 10 letters that are posted does not reach its destination. If Sanjay does not receive a reply to his letter, then what is the probability that Kesari did not receive Sanjay’s letter ? It is certain that Kesari will definitely reply to Sanjay’s letter if he receives it.
Answer: A
The probability is \(\frac{10}{19}\).
Sanjay gets no reply in two ways: his letter is lost, or it arrives and Kesari's reply is lost. Each letter is lost with probability 1/10.
\[ \begin{aligned} P(\text{no reply}) &= \tfrac{1}{10} + \tfrac{9}{10} \times \tfrac{1}{10} \\ &= \tfrac{19}{100} \\ P(\text{lost} \mid \text{no reply}) &= \tfrac{10/100}{19/100} \\ &= \tfrac{10}{19} \end{aligned} \]
\(\frac{1}{10}\) is the chance of loss before Sanjay knows there was no reply; the missing reply makes loss more likely. \(\frac{9}{100}\) is the chance that the letter arrived but the reply was lost, and \(\frac{9}{10}\) is the chance that a letter arrives.
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