Question

A policeman fires $6$ bullets at a burglar. The probability that the burglar will be hit by a bullet is $0.6$. What is the probability that the burglar is still unhurt?

Answer 1

Here we are given that, a policeman fires $6$ bullets at a burglar. So here we have to use Bernoulli’s trial. We have $n=6$ and $p=0.6$.

Complete step-by-step answer:
Now we are given that a policeman fires $6$ bullets at a burglar. The probability that the burglar will be hit by a bullet is $0.6$.
We have to use Bernoulli’s trial. The formula for Bernoulli’s trial is $P(\text{succes=x})={}^{n}{{C}_{x}}{{p}^{x}}{{q}^{n-x}}$.
Now here $x=0,1,2........,n$ and $p=0.6$ . Also, $q=1-p=1-0.6=0.4$ and $n=6$.
So, the probability that the burglar is still unhurt$=$ ${}^{6}{{C}_{0}}{{(0.6)}^{0}}{{(0.4)}^{6-0}}$
The probability that the burglar is still unhurt$=$ ${{(0.4)}^{6}}$
Simplifying we get,
The probability that the burglar is still unhurt$=$ $0.004096$
Therefore, the probability that the burglar is still unhurt is $0.004096$.

Additional information:
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes. Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Note: Bernoulli trial is also said to be a binomial trial. Many random experiments that we carry have only two outcomes that are either failure or success. Bernoulli’s trial formula is $P(\text{succes=x})={}^{n}{{C}_{x}}{{p}^{x}}{{q}^{n-x}}$.