Question

The probability of a shooter hitting a target is \( \frac{3}{4} \). Find the probability of hitting the target exactly 4 times in 5 shots.

• A. \(\frac{405}{1024}\)
• B. \(\frac{243}{1024}\)
• C. \(\frac{405}{512}\)
• D. \(\frac{81}{256}\)

Hint:

For “exactly \(r\)” successes in repeated independent trials, always apply the binomial formula \( {n \choose r}p^r q^{n-r} \).

Answer 1

The Correct Option is A

Concept: This problem follows the Binomial Distribution. If probability of success = \(p\) and probability of failure = \(q\), then probability of exactly \(r\) successes in \(n\) trials is: \[ P(X=r) = {n \choose r} p^r q^{\,n-r} \]

Step 1: Identify the parameters. \[ n = 5,\quad r = 4 \] \[ p = \frac{3}{4}, \quad q = 1 - p = \frac{1}{4} \]

Step 2: Apply the binomial probability formula. \[ P(X=4) = {5 \choose 4} \left(\frac{3}{4}\right)^4 \left(\frac{1}{4}\right) \]

Step 3: Simplify the expression. \[ {5 \choose 4} = 5 \] \[ P(X=4) = 5 \times \frac{81}{256} \times \frac{1}{4} \] \[ = 5 \times \frac{81}{1024} \] \[ = \frac{405}{1024} \] Thus, \[ \boxed{\frac{405}{1024}} \]