Question

1. on a january day in michigan, the weather channel predicts that there is a 70% chance of snow accumulations of at least 12 inches overnight. if at least 12 inches of snow accumulates, there is a 65% chance of a snow day on the following day. if less than 12 inches of snow accumulates, there is an 8% chance of a snow day.

a. make a tree diagram to model this chance process.
b. what is the probability that there is a snow day tomorrow?
c. if there was a snow day, what is the probability that there were snow accumulations of at least 12 inches?

Explanation:

Step1: Calculate the probability of snow day

Let \(A\) be the event that snow accumulations of at least 12 inches, \(P(A)=0.7\), then \(P(\overline{A}) = 1 - 0.7=0.3\). Let \(B\) be the event of a snow day.
If \(A\) occurs, \(P(B|A)=0.65\); if \(\overline{A}\) occurs, \(P(B|\overline{A}) = 0.08\)
By the law of total probability \(P(B)=P(A)P(B|A)+P(\overline{A})P(B|\overline{A})\)

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Step2: Calculate the conditional probability

By Bayes' theorem \(P(A|B)=\frac{P(A)P(B|A)}{P(B)}\)
Since \(P(A)P(B|A) = 0.7\times0.65=0.455\) and \(P(B) = 0.479\)

$$P(A|B)=\frac{0.455}{0.479}\approx0.9499$$

Answer:

b. The probability that there is a snow day tomorrow is \(0.479\)
c. If there was a snow day, the probability that there were snow accumulations of at least 12 inches is approximately \(0.95\)