Question

In a series of 3 independent trials, the probability of exactly 2 success is 12 times as large as the probability of 3 successes. The probability of a success in each trail is

• A. $\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is A $\frac{1}{5}$

For a binomial experiment consisting of n trials, the probability of exactly

k successes is

$P\left(k\text{successes}\right){=}^n{C}_k{p}^k(1-p{)}^{n-k}$

where the probability of success on each trial is p.

According to the question,

$n=3,,P\left(\text{2 successes}\right)=12\times P\left(\text{3 successes}\right)$

To find:

The probability of a success in each trail, p

$P\left(\text{2 successes}\right)=12\times P\left(\text{3 successes}\right)⟹{}^3{C}_2{p}^2(1-p{)}^{3-2}=12{\times }^3{C}_3{p}^3(1-p{)}^{3-3}⟹3\times p^2(1-p{)}^1=12\times 1\times p^3(1-p{)}^0⟹3p^2-3p^3=12p^3$

Divide throughout by $3p^2$, we get

$1-p=4p⟹5p=1⟹p=\frac{1}{5}$