Page 36 # 9 A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?
We have:
\(P(Art) = 5/8,P(French)=5/8, P(Art \cap French)=1/4\)
The Student must select exactly 2 out of the 3 electives
To find \(P(Math)\) we will have to find the probability of not choosing both Art and French since if both of those are chosen this means math was not.
So, since we know \(P(Art \cap French)\) we just need to compute \(P(Math) = 1 - P(Art \cap French)\)
\(P(Math) = 1 - 1/4\)
\(P(Math)=3/4\)
Then to compute \(P(Art \cup French)\) we need to compute \(P(Art \cup French) = P(Art) +P(French) - P(Art \cap French)\)
Therefore,
\(P(Art \cup French) = 5/8 + 5/8 - 1/4\)
\(P(Art \cup French) = 10/8 - 2/8\)
\(P(Art \cup French) =1\)
This makes sense because if we just think about the problem logically we can see that since there are only 3 electives to choose 2 from this means that we will always have to pick at least one of Art and French so therefore the probability is 1.