In 5 trials of binomial distribution, the probability of 3 successes is 4 times the probability of 2 successes. The probability of success in each trial is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{4}{5}\)
Let p be the probability of success and q = 1 − p. For n = 5, the binomial probability of r successes is \(P(X = r) = {}^{5}C_{r}\, p^r q^{5-r}\).
\(P(X = 3) = {}^{5}C_{3}\, p^3 q^2 = 10p^3q^2\)
\(P(X = 2) = {}^{5}C_{2}\, p^2 q^3 = 10p^2q^3\)
Given \(P(X = 3) = 4P(X = 2)\):
\(10p^3q^2 = 40p^2q^3\)
Dividing by \(10p^2q^2\) (p and q are not 0):
\(p = 4q = 4(1 - p)\)
\(5p = 4\)
\( \displaystyle p = \frac{4}{5} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{5}\) — Incorrect. This is the answer when the condition is reversed, \(P(X = 2) = 4P(X = 3)\), which gives q = 4p.
Option (3) \(\displaystyle\frac{3}{4}\) — Incorrect. Here \(\frac{p}{q} = 3\), so \(P(X = 3)\) would be 3 times \(P(X = 2)\), not 4 times.
Option (4) \(\displaystyle\frac{2}{3}\) — Incorrect. Here \(\frac{p}{q} = 2\), so the ratio would be 2, not 4.