Question

find the probability exactly 3 successes in 6 trials of a binomial experiment in which the probability of success if 50%. round to the nearest tenth of a percent.

Answer 1

Hence, the probability of exactly 3 successes in 6 trials of a binomial experiment round to the nearest tenth of a percent is:

31.2%

Explanation:

The probability of getting exactly k successes in n trials is given by the probability mass function:

`{\displaystyle P(k;n,p)=P(X=k)={\binom {n}{k}}p^(k)(1-p)^(n-k)}`

Where p denotes the probability of success.

We are given that the probability of success if 50%.

i.e.
`p=(1)/(2)`

also form the question we have:

k=3 and n=6.

Hence the probability of exactly 3 successes in 6 trials is:

`{\displaystyle P(3;6,(1)/(2))=P(X=3)={\binom {6}{3}}((1)/(2))^(3)(1-(1)/(2))^(6-3)}`

`{\displaystyle P(3;6,(1)/(2))=P(X=3)={\binom {6}{3}}((1)/(2))^(3)((1)/(2))^(3)}`

`{\displaystyle P(3;6,(1)/(2))=P(X=3)={\binom {6}{3}}((1)/(2))^(6)`

`\binom {6}{3}=20`

Hence,

`{\displaystyle P(3;6,(1)/(2))=P(X=3)=20* ((1)/(2))^6=(5)/(16)`

In percentage the probability will be:

`(5)/(16)* 100=31.25\%=31.2\%`