Question

A man succeeds twice as often as he fails. What is the probability that in the next 5 trials there will be four successes?

• A. 5*(2/3)*(1/3)⁴
• B. (2/3)*(1/3)⁴
• C. 5*(2/3)⁴*(1/3)
• D. (2/3)⁴*(1/3)
• E. None of these

Answer 1

The correct option is C

5*(2/3)⁴*(1/3)

An experiment succeeds twice as often as it fails.

i.e. the probability of its success is $\frac{2}{3}$ and the probability of its failure is $\frac{1}{3}$.

In the next 5 trials, the experiment needs to succeed in 4 out of the 5 trials.

4 out of the 5 trials in which it succeeds could be selected in ${}^5{C}_4$ = 5 ways.

And as 4 of them are successes, they have a probability of $\frac{2}{3}$ and the one that is a failure will have a probability of $\frac{1}{3}$.

Hence, the required probability = 5*($\frac{2}{3}$)${}^4$*($\frac{1}{3}$).