In general, the probability that it rains on Saturday is 25%. If it rains on Saturday, the probability that it rains on Sunday is 50%. If it does not rain on Saturday, the probability that it rains on Sunday is 25%. Given that it rained on Sunday, what is the probability that it rained on Saturday?

Answer:

40%

Step-by-step explanation:

From the given statements:

The probability that it rains on Saturday is 25%.

P(Sunday)=25%=0.25

Given that it rains on Saturday, the probability that it rains on Sunday is 50%.

P(Sunday|Saturday)=50%=0.5

Given that it does not rain on Saturday, the probability that it rains on Sunday is 25%.

P(Sunday|No Rain on Saturday)=25%=0.25

We are to determine the probability that it rained on Saturday given that it rained on Sunday, P(Saturday|Sunday).

P(No rain on Saturday)=1-P(Saturday)=1-0.25=0.75

Using Bayes Theorem for conditional probability:

P(Saturday|Sunday)=[TeX]\frac{P(Sunday|Saturday)P(Saturday)}{P(Sunday|Saturday)P(Saturday)+P(Sunday|No Rain on Saturday)P(No Rain on Saturday)}[/TeX]

=[TeX]\frac{0.5*0.25}{0.5*0.25+0.25*0.75}[/TeX]

=0.4

There is a 40% probability that it rained on Saturday given that it rains on Sunday.

Final answer:

To find the probability that it rained on Saturday given that it rained on Sunday, we can use Bayes' theorem. We are given the probabilities of rain on Saturday and Sunday, and we can use the law of total probability to calculate the probability of rain on Sunday. Then, using Bayes' theorem, we can determine the probability of rain on Saturday given that it rained on Sunday.

Explanation:

We need to use Bayes' theorem to find the probability that it rained on Saturday given that it rained on Sunday. Let's denote R1 as the event that it rains on Saturday and R2 as the event that it rains on Sunday. We are given P(R1) = 0.25, P(R2|R1) = 0.50, and P(R2|~R1) = 0.25, where ~R1 represents the event that it does not rain on Saturday. We want to find P(R1|R2), which is the probability that it rained on Saturday given that it rained on Sunday.

1. First, let's find P(R2).
2. Using the law of total probability, we can express P(R2) as P(R2|R1)P(R1) + P(R2|~R1)P(~R1).
3. Since P(R2|R1) = 0.50, P(R1) = 0.25, P(R2|~R1) = 0.25, and P(~R1) = 1 - P(R1) = 0.75, we can substitute these values into the equation and calculate P(R2).
4. Next, we can use Bayes' theorem to find P(R1|R2).
5. Bayes' theorem states that P(R1|R2) = (P(R2|R1)P(R1))/P(R2).
6. Substituting the values we know, we get P(R1|R2) = (0.50*0.25)/P(R2).
7. We can use the value we calculated for P(R2) in the previous step to find P(R1|R2).

Calculating these values will give us the probability that it rained on Saturday given that it rained on Sunday.

Learn more about Bayes' theorem here:

brainly.com/question/29598596

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