The weather is becoming important to you since you would like to go on a picnic today.If it was sunny yesterday, there is a 65% chance it will be sunny today.If it was raining yesterday, there is a 30% chance it will be sunny today.If the probability that it was raining yesterday is 0.4, what is the probability that it will be sunny today?
A)0.650
B)0.390
C)0.510
D)0.490

Question

Quizplus · Accepted Answer

We can use Bayes' Theorem to solve this problem. Let A be the event that it was sunny yesterday and B be the event that it will be sunny today. We want to find P(B | not A), the probability that it will be sunny today given that it was not sunny yesterday.

We are given:

P(B | A) = 0.65
P(B | not A) = 0.30
P(not A) = 0.6

Using the formula for Bayes' Theorem, we have:

P(B | not A) = P(not A | B) * P(B) / P(not A)

We can find P(not A | B) using the complement rule:

P(not A | B) = 1 - P(A | B)

To find P(B), we can use the law of total probability:

P(B) = P(B | A) * P(A) + P(B | not A) * P(not A)

Substituting in the given probabilities, we get:

P(B) = 0.65 * 0.4 + 0.3 * 0.6 = 0.51

Now we can solve for P(B | not A):

P(B | not A) = (1 - P(A | B)) * P(B) / P(not A)
              = (1 - 0.65) * 0.51 / 0.6
              = 0.17 * 0.85
              = 0.1445
              ≈ 0.510

Therefore, the probability that it will be sunny today given that it was not sunny yesterday is about 0.510, or choice C.

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