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

Answer:

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%

Step-by-step 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. $p=\dfrac{1}{2}$

also form the question we have:

k=3 and n=6.

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

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

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

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

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

Hence,

${\displaystyle P(3;6,\dfrac{1}{2})=P(X=3)=20\times (\dfrac{1}{2})^6=\dfrac{5}{16}$

In percentage the probability will be:

$\dfrac{5}{16}\times 100=31.25\%=31.2\%$