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Sati [7]
3 years ago
9

Two roommates share a dorm in Richards Hall, The probability that Roommate A is doing homework on a Friday night is .3. The prob

ability that Roommate B is doing homework on a Friday night is .4. The probability that both roommates are doing homework on a Friday night is .2 (notice the two roommates are able to coordinate their decisions, so you can't assume their respective probabilities are independent). Find the probability that: At least one roommate is doing homework this Friday night
Mathematics
1 answer:
bixtya [17]3 years ago
4 0

Answer:

There is a 50% probability that at least one roommate is doing homework this Friday night.

Step-by-step explanation:

This problem can be solved building the Venn Diagram of these probabilities.

I am going to say that P(A) is the probability that the roommate A is doing homework and P(B) is the probability that the roommate B is doing homework.

We have that:

P(A) = P(a) + P(A \cap B)

In which P(a) is the probability that only the roommate A is doing homework and P(A \cap B) is the probability that both student A and student B are doing homework.

We also have that:

P(B) = P(b) + P(A \cap B)

The problem states that

The probability that Roommate A is doing homework on a Friday night is .3. So P(A) = 0.3.

The probability that Roommate B is doing homework on a Friday night is .4. So P(B) = 0.4

The probability that both roommates are doing homework on a Friday night is .2. So P(A \cap B) = 0.2

Find the probability that: At least one roommate is doing homework this Friday night

This is the probability that either only A is doing, either only B, or both. So:

P = P(a) + P(b) + P(A \cap B)

We have that

P(A) = P(a) + P(A \cap B)

We have P(A) and P(A \cap B), so we can find P(a)

P(A) = P(a) + P(A \cap B)

0.3 = P(a) + 0.2

P(a) = 0.1

Also

P(B) = P(b) + P(A \cap B)

0.4 = P(b) + 0.2

P(b) = 0.2

So:

P = P(a) + P(b) + P(A \cap B)

P = 0.1 + 0.2 + 0.2

P = 0.5

There is a 50% probability that at least one roommate is doing homework this Friday night.

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