Question

The probability that it will rain tomorrow is 1/4 . The probability that Chris will be late to his office given that it does not rain tomorrow is 5/6 and the probability that Chris will be late to his office given that it rains tomorrow is 2/ 3 . What is the probability that it will rain tomorrow, given that Chris is not late to his office?

Answer 1

Using Bayes' Theorem, the probability that it will rain tomorrow, given that Chris is not late to his office, is 2/7.

Step-by-step explanation:

To find the probability that it will rain tomorrow, given that Chris is not late to his office, we can use Bayes' Theorem. Bayes' Theorem states that P(A|B) = P(B|A) * P(A) / P(B), where A and B are events. In this case, event A is rain tomorrow and event B is Chris not being late to his office.

We are given P(A) = 1/4, P(B|A') = 5/6, and P(B|A) = 2/3, where A' represents the complement of event A (i.e., no rain tomorrow).

First, let's find P(B) using the law of total probability. P(B) = P(B|A) * P(A) + P(B|A') * P(A') = (2/3) * (1/4) + (5/6) * (3/4) = 7/12.

Now, substituting the values into Bayes' Theorem: P(A|B) = (2/3) * (1/4) / (7/12) = 2/7.

Answer 2

To determine the probability that it will rain tomorrow, given that Chris is not late to his office, we utilize Bayes' theorem or a probability tree, taking into account the given probabilities of rain and Chris being late. The calculation leads to the result that the desired probability is 1/9.

Step-by-step explanation:

The question asks to find the probability that it will rain tomorrow, given that Chris is not late to his office. This is a conditional probability problem that can be solved using Bayes' theorem or a probability tree. The problem provides the probability of rain (1/4) and the probabilities that Chris will be late to his office given that it does or does not rain. To find the desired probability, we need to consider the likelihood of rain when we know Chris is on time. Let's denote:

R as the event that it rains.

L as the event that Chris is late.

R' as the event that it does not rain.

L' as the event that Chris is not late.

The provided information translates to:

P(R) = 1/4 (Probability of rain)

P(L|R') = 5/6 (Probability Chris is late if it does not rain)

P(L|R) = 2/3 (Probability Chris is late if it does rain)

Since the probability of Chris being late is the sum of him being late whether it rains or not:

P(L) = P(L|R) * P(R) + P(L|R') * P(R')

We can find P(R') = 1 - P(R) = 3/4. Plugging in the numbers, we get:

P(L) = (2/3) * (1/4) + (5/6) * (3/4) = 1/6 + 15/24 = 5/8

The probability that Chris is not late is then P(L') = 1 - P(L) = 3/8. Now, we apply Bayes' theorem to find P(R|L'), the probability that it rains given Chris is not late:

P(R|L') = [P(L'|R) * P(R)] / P(L')

To find P(L'|R), we use the complement of P(L|R), which is P(L'|R) = 1 - P(L|R) = 1/3.

Finally, plugging in the numbers:

P(R|L') = (1/3) * (1/4) / (3/8) = (1/12) / (3/8) = 1/9

Therefore, the probability that it will rain tomorrow, given that Chris is not late to his office, is 1/9.