Question

There is a 30% chance that T'Shana's county will have a drought during any given year. She performs a

simulation to find the experimental probability of a drought in at least 1 of the next 4 years

Answer 1

75.99% probability of a drought in at least 1 of the next 4 years

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.

`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.

30% chance that T'Shana's county will have a drought during any given year.

This means that
`p = 0.3`

4 years

This means that
`n = 4`

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

`P(X = 0) + P(X \geq 1) = 1`

We want
`P(X \geq 1)`. So

`P(X \geq 1) = 1 - P(X = 0)`

In which

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

`P(X = 0) = C_(4,0).(0.3)^(0).(0.7)^(4) = 0.2401`

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2401 = 0.7599`

75.99% probability of a drought in at least 1 of the next 4 years