Question

you are planning a three - day trip to seattle, washington, in october. use the fact that on each day, it could either be sunny or rainy, and that each day is equally likely to be sunny or rainy to answer the following question.
what is the probability that it rains all three days?
(type an integer or a decimal. do not round.)

Explanation:

Step1: Determine the probability of rain on a single day

Since each day is equally likely to be sunny or rainy, the probability of rain on a single day \( P(\text{rain})=\frac{1}{2}\)

Step2: Use the multiplication rule for independent events

The events (rain on day 1, rain on day 2, rain on day 3) are independent. For independent events \(A\), \(B\), and \(C\), \(P(A\cap B\cap C)=P(A)\times P(B)\times P(C)\)
Let \(A\), \(B\), and \(C\) be the events of rain on day 1, day 2, and day 3 respectively. Then \(P(A) = P(B)=P(C)=\frac{1}{2}\)
So \(P(\text{rain all 3 days})=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\)

Step3: Calculate the product

\(\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1\times1\times1}{2\times2\times2}=\frac{1}{8} = 0.125\)

Answer:

\(0.125\)