Question

The probability of A hitting a target is $\dfrac{2}{3}$ and that of B is $\dfrac{4}{5}$. They both fire at the target. What is the probability that at best one of them hits the target? What is the probability that only one of them will hit the target?

Answer 1

Here, we are given the probability of A and B when hitting the target. We need to find the probability that one of them hits the target means that probability both hits the target subtracted from one. And next, we need to find the probability that only one of them will hit the target means that the probability of A XOR the probability of B. On solving this, we will get the final output.

Complete step-by-step answer:
Given that,
The probability of A hitting a target: $P(A) = \dfrac{2}{3}$ and the probability of B hitting a target: $P(B) = \dfrac{4}{5}$.
So, the probability of not hitting the target:
$P(A’) = 1 - \dfrac{2}{3}$
$= \dfrac{{3 - 2}}{3}$
$= \dfrac{1}{3}$

Now, we will find the probability that at best one of them hits the target is:
= 1 – the probability that both hits the target
$= 1 - P(A)P(B)$
Substituting the values, we will get,
$= 1 - \left( {\dfrac{2}{3}} \right)\left( {\dfrac{4}{5}} \right)$
$= 1 - \dfrac{8}{{15}}$
Taking LCM as 15, we will get,
$= \dfrac{{15 - 8}}{{15}}$
On simplifying this, we will get,
$= \dfrac{7}{{15}}$

Thus, the probability that one of them will hit the target is $\dfrac{7}{{15}}$ .

Next, we will find the probability that one of them will hit the target is:
= the probability that A hits the target XOR the probability that B hits the target
= P(A) + P(B) – 2P(A AND B both hits)
= P(A) + P(B) – 2P(A)P(B)
Substituting the values, we will get,
$= \dfrac{2}{3} + \dfrac{4}{5} - 2\left( {\dfrac{2}{3}} \right)\left( {\dfrac{4}{5}} \right)$
$= \dfrac{2}{3} + \dfrac{4}{5} - \dfrac{{16}}{{15}}$
Taking LCM as 15, we will get,
$= \dfrac{{2(5) + 4(3) - 16}}{{15}}$
On evaluating this, we will get,
$= \dfrac{{10 + 12 - 16}}{{15}}$
$= \dfrac{{22 - 16}}{{15}}$
$= \dfrac{6}{{15}}$
$= \dfrac{2}{5}$

Thus, the probability that one of them will hit the target is $\dfrac{2}{5}$ .

Hence, the probability that at best one of them hits the target is $\dfrac{7}{{15}}$ and the probability that one of them will hit the target is $\dfrac{2}{5}$ .

Note: Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. The formulas of probability are:

1. AND: P(A and B) = P(A).P(B)
2. OR: P(A or B) = P(A) + P(B) – P(A and B)
3. XOR: P(A xor B) = P(A) + P(B) – 2P(A and B)