Question

What is the probability of `X= 6` successes, using the binomial formula?

Let `n=13, p = .43`.

Answer 1

`P(X=6) ~~ 0.2121`, or approximately `21.21%`.

Explanation:

The binomial formula states that the probability of getting exactly `x` successes out of `n` trials (where each independent trial has success probability `p`) is

`P(X=x)=((n),(x))p^x(1-p)^(n-x)," "x=0,1,2,...,n`

where

• `((n),(x))` is the number of ways to position `x` successes in a sequence of `n` trials, equal to `(n!)/(x!(n-x)!)`;
• `p^x` is the probability of getting those `x` independent successes; and
• `(1-p)^(n-x)` is the probability of failure for the remaining `n-x` independent trials.

To obtain an answer, we simply plug in the given values of `n`, `p`, and `x`:

`P(X=6)=((13),(6))(.43)^6(1-.43)^(13-6)`

`color(white)(P(X=6))=(13!)/(6!(13-6)!)(.43)^6(.57)^7`

`color(white)(P(X=6)) ~~ (1716)(0.006321)(0.015949)`

`color(white)(P(X=6)) ~~ 0.2121`

So, out of 13 trials, the probability of obtaining exactly 6 successes is

`P(X=6) ~~ 0.2121`
`color(white)(P(X=6)) = 21.21%`.