Question

Use this information for Items 5–7. There is a 60% probability of rain each of the next 5 days. Find each probability. Round to the nearest whole percent. PLEASE HELP ME!!!​

Answer 1

0.6808 = 68.08% probability that it will rain on at least 3 of the next 5 days.

0.9898 = 98.98% probability it will rain on at least 1 of the next 5 days.

0.84 = 84% probability it will rain on at least 1 of the next 2 days.

Explanation:

For each day, there are only two possible outcomes. Either it rains, or it does not. The probability of raining on a day is independent of any other day. This means that 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.

There is a 60% probability of rain each of the next 5 days.

This means that
`p = 0.6, n = 5`

Probability it will rain on at least 3 of the next 5 days:

This is:

`P(X \geq 3) = 1 - P(X < 3)`

In which

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

So

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

`P(X = 0) = C_(5,0).(0.6)^(0).(0.4)^(5) = 0.0102`

`P(X = 1) = C_(5,1).(0.6)^(1).(0.4)^(4) = 0.0768`

`P(X = 2) = C_(5,2).(0.6)^(2).(0.4)^(3) = 0.2304`

So

`P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0120 + 0.0768 + 0.2304 = 0.3192`

`P(X \geq 3) = 1 - P(X < 3) = 1 - 0.3192 = 0.6808`

0.6808 = 68.08% probability that it will rain on at least 3 of the next 5 days.

Probability it will rain on at least 1 of the next 5 days:

This is:

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

From the above item, we have that
`P(X = 0) = 0.0102`. So

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0102 = 0.9898`

0.9898 = 98.98% probability it will rain on at least 1 of the next 5 days.

Probability it will rain on at least 1 of the next 2 days:

Two days means that
`n = 2`.

The probability is:

`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_(2,0).(0.6)^(0).(0.4)^(2) = 0.16`

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.16 = 0.84`

0.84 = 84% probability it will rain on at least 1 of the next 2 days.