Question

if the probability of hitting a target is 3/4 and 9 shots are fired independently, what is the probability of hitting the target once or not at all?

Answer 1

If the probability of hitting a target is $3/4$ and 9 shots are fired independently, what is the probability of hitting the target once or not at all?

Solution:

Let the random variable $X$ corresponds to the number of hits of a target out of 9 shots.

Here $X \sim B(n, p)$, where

$n = 9$ - number of shots which are fired independently

$p = P(\text{hitting a target}) = \frac{3}{4} = 0.75$

$$P(X = r) = P(r) = {}^n C_r p^r q^{n-r}$$ $$q = 1-p$$ $$r = 0, 1, 2, 3, \dots, 9$$ $$P(X = r) = P(r) = {}^9 C_r (0.75)^r (1 - 0.75)^{9-r} = {}^9 C_r (0.75)^r (0.25)^{9-r}$$

Required probability:

$$P(X \geq 1) = 1 - P(X = 0) = 1 - {}^9 C_0 (0.75)^0 (0.25)^9 = 1 - (0.25)^9 = 0.9999962$$ $$P(X = 0) = 0.0000038$$

Answer:

The probability of hitting the target once is 0.9999962.

The probability of hitting the target not at all is 0.0000038.