Question

Find the probability of exactly 5 successes in 7 trials of a binomial experiment in which the probability of success is 70%

Answer 1

`\blue{\huge {\mathrm{PROBABILITY}}}`

`\\`

`{===========================================}`

`{\underline{\huge \mathbb{Q}{\large \mathrm {UESTION : }}}}`

• Find the probability of exactly 5 successes in 7 trials of a binomial experiment in which the probability of success is 70%.

`{===========================================}`

`{\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}}`

• The probability of exactly 5 successes in 7 trials of a binomial experiment in which the probability of success is 70% is 0.3176523.

`{===========================================}`

`{\underline{\huge \mathbb{S} {\large \mathrm {OLUTION : }}}}`

We can use the binomial probability formula to find the probability of exactly 5 successes in 7 trials:

`\sf P(5\: successes\: in\: 7\: trials) = (7\: choose\: 5) (0.7)^5 (0.3)^2`

where:

• n = 7 (number of trials),
• p = 0.7 (probability of success),
• q = 0.3 (probability of failure), and
• (7 choose 5) = 7!/(5!2!) = 21 (the number of ways to choose 5 successes in 7 trials).

Plugging in these values, we get:

`\begin{aligned}\sf P(5\: successes\: in\: 7\: trials)& =\sf 21 (0.7)^5 (0.3)^2\\& =\bold{ 0.3176523}\end{aligned}`

Therefore, the probability of exactly 5 successes in 7 trials of a binomial experiment in which the probability of success is 70% is 0.3176523.

`{===========================================}`

`- \large\sf\copyright \: \large\tt{AriesLaveau}\large\qquad\qquad\qquad\tt 04/01/2023`

Answer 2

0.3176523

Explanation:

You want the probability of exactly 5 successes in 7 trials of a binomial experiment in which the probability of success is 70%.

Probability

The desired probability is the product of the probability of 5 success and 2 failures, multiplied by the number of ways that result might occur.

P(x=5) = 7C5·(0.7^5)(1 -0.7)^2 = 0.3176523

The probability of exactly 5 successes is 0.3176523.

__

Additional comment

Any of a number of probability calculators can tell you this probability. Spreadsheets have appropriate functions built in as well.