Answer:
99.8% probability of at least one failure.
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.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Probability of success is 30%.
This means that [tex]p = 0.3[/tex]
Five trials:
This means that [tex]n = 5[/tex]
Find the probability of at least one failure in five trials of a binomial experiment in which the probability of success is 30%.
Less than five sucesses, which is:
[tex]P(X < 5) = 1 - P(X = 5)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{5,5}.(0.3)^{5}.(0.7)^{0} = 0.002[/tex]
[tex]P(X < 5) = 1 - P(X = 5) = 1 - 0.002 = 0.998[/tex]
0.998*100% = 99.8%
99.8% probability of at least one failure.