Question

During World War II, the British found that the probability that a bomber is lost through enemy action on a mission over occupied Europe was $0.05$ Assuming that missions are independent, find the probability that a bomber returned safely from $20$ missions.

Short answer

The probability is $0.3585$

Step-by-step solution

Step 1. Given Information

The chance of the bomber being lost is $0.05$ The missions are self- contained.

Step 2. Concept Used

Two mutually exclusive events can never be independent since if one happens, the other is nearly surely going to happen as well.

Step 3. Calculation

The bomber's safety is calculated as follows:$$\begin{gathered} P\left(Safe\right)=1-P\left(lost\right) \\ P\left(Safe\right)=1-0.05=0.95 \end{gathered}$$

After $20$trips, the probability that the bomber will return safely is computed as follows:$P(Willcomebacksafely)={0.95}^{20}=0.3585$

As a result, the necessary probability is $0.3585$