The ratio of probability of 3 successes in 5 independent trials to the probability of 2 successes in 5 independent trials is $\frac{1}{4}$. What is the probability of 4 successes in 6 independent trials.

Subject: Applied Mathematics 4

Topic: Probability

Difficulty: Medium

$ \frac{P(X=3)}{P(X=2)} = \frac{1}{4} $

n = 5; q = 1 - p

$ \frac{^5C_3p^3q^{5-3}}{^5C_2p^2q^{5-2}} = \frac{1}{4} \\ \frac{10 p^3q^2}{10p^2q^3} = \frac{1}{4} \\ \frac{p}{q} = \frac{1}{4} \\ \frac{p}{1-p} = \frac{1}{4} \implies 4p = 1 - p \implies p = \frac{1}{5} \therefore q = \frac{4}{5} $

$ P(X=r) = ^nC_rp^rq^{n-r} \\ = \, ^6C_r (\frac{1}{5})^r (\frac{4}{5})^{6-r} \\ P(X=4) = \, ^6C_4 (\frac{1}{5})^4 (\frac{4}{5})^{6-4} \\ = \, ^6C_4 (\frac{1}{5})^4 (\frac{4}{5})^{2} \\ = \, ^6C_4 \times \frac{4^2}{5^6} \\ = 15 \times \frac{16}{15625} = \frac{48}{3125} $