3. on a january day in michigan, the weather channel predicts that there is a 70% chance of snow…

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

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

Answer

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})) [ \begin{align*} P(B)&=0.7\times0.65 + 0.3\times0.08\ &=0.455+0.024\ &=0.479 \end{align*} ]

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)