Question

A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of x successes in the n independent trials of the experiment.

n = 8​, p = 0.3​, x <4
a. P(X < 4) = ___________

Answer 1

The probability of getting less than 4 successes in 8 independent trials of a binomial probability experiment with a success probability of 0.3 is approximately 0.789.

Step-by-step explanation:

The probability of getting less than 4 successes in 8 independent trials of a binomial probability experiment with a success probability of 0.3 can be found using technology.

Using a binomial calculator or software, calculate the probability of getting 0, 1, 2, or 3 successes in 8 trials.

Add these probabilities together to find the probability of getting less than 4 successes.

For P(X < 4), using technology, the probability is approximately 0.789.

Answer 2

P(X < 4) = 0.8059

Step-by-step explanation:

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

In this problem we have that:

`n = 8, p = 0.3`

`P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(8,0).(0.3)^(0).(0.7)^(8) = 0.0576`

`P(X = 1) = C_(8,1).(0.3)^(1).(0.7)^(7) = 0.1977`

`P(X = 2) = C_(8,2).(0.3)^(2).(0.7)^(6) = 0.2965`

`P(X = 3) = C_(8,3).(0.3)^(3).(0.7)^(5) = 0.2541`

`P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0576 + 0.1977 + 0.2965 + 0.2541 = 0.8059`

P(X < 4) = 0.8059