Respuesta :
Answer:
75.99% probability of a drought in at least 1 of the next 4 years
Step-by-step explanation:
For each year, there are only two possible outcomes. Either there is a drought, or there is not. The probability of there being a drought in a given year is independent of other years. So the binomial probability distribution is used to solve this question.
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.
30% chance that T'Shana's county will have a drought during any given year.
This means that [tex]p = 0.3[/tex]
4 years
This means that [tex]n = 4[/tex]
She performs a
She performs a simulation to find the experimental probability of a drought in at least 1 of the next 4 years
Either no year has a drought, or at least one has. The sum of the probabilities of these events is decimal 1. So
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
We want [tex]P(X \geq 1)[/tex]. So
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{4,0}.(0.3)^{0}.(0.7)^{4} = 0.2401[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2401 = 0.7599[/tex]
75.99% probability of a drought in at least 1 of the next 4 years
