On each of three consecutive days the National Weather Service announces that there is a 50-50 chance of rain. Assuming that the National Weather Service is correct, what is the probability that it rains on at most one of the three days? (Hint: Represent the outcome that it rains on day 1 and doesn’t rain on days 2 and 3 as RNN.)

1. 50%
2. 33%
3. 37.5%.
4. None of the above.

For each day, there are only two possible outcomes. Either it does rain, or it does not, so we use the binomial probability distribution to solve this problem.

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 combinatios 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.

In this problem we have that:

[tex]n = 3, p = 0.5[/tex]

We want to find

NNN + RNN + NRN + NNR

This is

[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]

So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{3,0}.(0.5)^{0}.(0.5)^{3} = 0.125[/tex]

[tex]P(X = 1) = C_{3,1}.(0.5)^{1}.(0.5)^{2} = 0.375/tex]

[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.125 + 0.375 = 0.5[/tex]

The correct answer is:

1. 50%