Find the probability of 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.

The probability of exactly 3 successes in 6 trials of a binomial experiment in which the probability of success if 50%, is:

31.3% (Rounding to the nearest tenth)

Step-by-step explanation:

With the information provided, P = 0.5, n = 6 and k = 3, we can build the binomial distribution table, this way:

Binomial distribution (n=6, p=0.5)

 f(x)F(x)

xPr[X = x]Pr[X ≤ x]

00.01560.0156

10.09380.1094  

20.23440.3438

30.31250.6563  

40.23440.8906  

50.09380.9844  

60.01561.0000

Under the first column we have the probabilities of success of every value from 0 to 6, and under the second column we have the values for answering the question at least how much probability we have of any number of successes. Our question is exactly 3 successes in 6 trials and we see that value under the first column: 0.3125.

Therefore, the probability of exactly 3 successes in 6 trials of a binomial experiment in which the probability of success if 50%, is:

31.3% (Rounding to the nearest tenth)

Let's recall that the formula for calculating the probability of exact successes is:

P(k out of n) =   (n!/k!(n-k)!) * (p∧k*(1-p)∧(n-k))

in our case p=0.5, n=6, k=3

(n!/k!(n-k)!) = 20 and (p∧k*(1-p)∧(n-k)) = 0.015625

20 * 0.015625 = 0.3125

picture below