Respuesta :
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 that
The 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 night
This 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.
