An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Question

Philoid · Accepted Answer

Let p be probability of success and q be probability of failure of experiment

Given that p = 3q

As we have only two possibilities success or failure hence the distribution is binomial

And in binomial distribution p + q = 1

⇒ 3q + q = 1

⇒ q = 1/4

Put q = 1/4 in p + q = 1

⇒ p + 1/4 = 1

⇒ p = 1 – 1/4

⇒ p = 3/4

Now for binomial distribution the probability P(X=x) is given by

Where n is the total number of trials and x is the required result

In 5 trials we require at least 3 success

Hence n = 5 and at least 3 means 3 success, 4 success or 5 success which means x = 3, 4 or 5

Hence required probability

⇒ required probability = P(X = 3) + P(X = 4) + P(X = 5)

Using

Hence the probability of at least 3 success in 5 trials is .