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

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:[tex]P(A) = P(a) + P(A \cap B)[/tex]In which P(a) is the probability that only the roommate A is doing homework and [tex]P(A \cap B)[/tex] is the probability that both student A and student B are doing homework.We also have that:[tex]P(B) = P(b) + P(A \cap B)[/tex]The problem states thatThe probability that Roommate A is doing homework on a Friday night is .3. So [tex]P(A) = 0.3[/tex].The probability that Roommate B is doing homework on a Friday night is .4. So [tex]P(B) = 0.4[/tex]The probability that both roommates are doing homework on a Friday night is .2. So [tex]P(A \cap B) = 0.2[/tex]Find the probability that: At least one roommate is doing homework this Friday nightThis is the probability that either only A is doing, either only B, or both. So:[tex]P = P(a) + P(b) + P(A \cap B)[/tex]We have that[tex]P(A) = P(a) + P(A \cap B)[/tex]We have P(A) and [tex]P(A \cap B)[/tex], so we can find P(a)[tex]P(A) = P(a) + P(A \cap B)[/tex][tex]0.3 = P(a) + 0.2[/tex][tex]P(a) = 0.1[/tex]Also[tex]P(B) = P(b) + P(A \cap B)[/tex][tex]0.4 = P(b) + 0.2[/tex][tex]P(b) = 0.2[/tex]So:[tex]P = P(a) + P(b) + P(A \cap B)[/tex][tex]P = 0.1 + 0.2 + 0.2[/tex][tex]P = 0.5[/tex]There is a 50% probability that at least one roommate is doing homework this Friday night.