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?

1. according to survey results from the pew research center, 30% of people made new years resolutions. of the 5,140 people who were surveyed, 16% were in the 18 - 29 age bracket. of the people who were surveyed, 7.9% were in the 18 - 29 age bracket and made new years resolutions. suppose that a person from the survey is randomly selected.

a. are the events \made a new years resolution\ and \18 - 29 years old\ independent? justify your answer.
b. is a person between the ages of 18 - 29 more likely, less likely, or equally likely to make a new years resolution than all survey responders? explain.

Explanation:

Step1: Calculate the probability of snow day

Let \(A\) be the event of snow accumulation \(\geq12\) inches, \(P(A) = 0.7\), and the probability of snow day given \(A\) is \(P(S|A)=0.65\). Let \(B\) be the event of snow accumulation \(< 12\) inches, \(P(B)=1 - 0.7=0.3\), and the probability of snow day given \(B\) is \(P(S|B) = 0.08\).
By the law of total probability \(P(S)=P(A)\times P(S|A)+P(B)\times P(S|B)\)

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Step2: Calculate the probability using Bayes' theorem

By Bayes' theorem \(P(A|S)=\frac{P(A)\times P(S|A)}{P(S)}\)
Since \(P(A) = 0.7\), \(P(S|A)=0.65\) and \(P(S) = 0.479\)

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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 \(0.95\).