<p>A die is thrown a fixed number of times. If probability of getting even number 3 times is same as the probability of getting even number 4 times, then probability of getting even number exactly once is</p>
Step-by-Step Solution
Key Concept: When P(X=3) = P(X=4) for a binomial distribution with p=1/2, use the equality condition to find n, then calculate P(X=1) using the same n.
<p><strong>Step 1:</strong> Set up the binomial probability equation. Let n = number of throws, p = 1/2 (probability of even). Given: P(X=3) = P(X=4)</p><p><strong>Step 2:</strong> Write the equality: $\binom{n}{3}\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^{n-3} = \binom{n}{4}\left(\frac{1}{2}\right)^4\left(\frac{1}{2}\right)^{n-4}$</p><p><strong>Step 3:</strong> Simplify: $\binom{n}{3} = \binom{n}{4}$</p><p>This gives: $\frac{n!}{3!(n-3)!} = \frac{n!}{4!(n-4)!}$</p><p><strong>Step 4:</strong> Cross multiply and solve: $4!(n-4)! = 3!(n-3)!$</p><p>$24(n-4)! = 6(n-3)(n-4)!$</p><p>$24 = 6(n-3)$ → $n = 7$</p><p><strong>Step 5:</strong> Find P(X=1) with n=7, p=1/2:</p><p>$P(X=1) = \binom{7}{1}\left(\frac{1}{2}\right)^7 = 7 \cdot \frac{1}{128} = \frac{7}{128}$</p><p>∴ Answer: D</p>
Correct Answer: D