Solution Found!
Suppose the events B1 and B2 are mutually exclusive and
Chapter 3, Problem 77E(choose chapter or problem)
Suppose the events \(B_{1}\) and \(B_{2}\) are mutually exclusive and complementary events, such that \(P\left(B_{1}\right)=.75\) and \(P\left(B_{2}\right)=.25\). Consider another event A such that \(P\left(A \mid B_{1}\right)=.3\) and \(P\left(A \mid B_{2}\right)=.5\).
a. Find \(P\left(B_{1} \cap A\right)\).
b. Find \(P\left(B_{2} \cap A\right)\).
c. Find P(A) using the results in parts a and b.
d. Find \(P\left(B_{1} \mid A\right)\).
e. Find \(P\left(B_{2} \mid A\right)\).
Questions & Answers
QUESTION:
Suppose the events \(B_{1}\) and \(B_{2}\) are mutually exclusive and complementary events, such that \(P\left(B_{1}\right)=.75\) and \(P\left(B_{2}\right)=.25\). Consider another event A such that \(P\left(A \mid B_{1}\right)=.3\) and \(P\left(A \mid B_{2}\right)=.5\).
a. Find \(P\left(B_{1} \cap A\right)\).
b. Find \(P\left(B_{2} \cap A\right)\).
c. Find P(A) using the results in parts a and b.
d. Find \(P\left(B_{1} \mid A\right)\).
e. Find \(P\left(B_{2} \mid A\right)\).
ANSWER:Step 1 of 5
Let the events \(B_{1}\) and \(B_{2}\) are mutually exclusive and complementary events.
So \(P\left(B_{1}\right)=0.75\) and \(P\left(B_{2}\right)=0.25\).
We consider another \(P\left(A / B_{1}\right)=0.30\) and \(P\left(A / B_{2}\right)=0.50\).
Our goal is:
a). We need to find \(P\left(B_{1} \cap A\right)\).
b). We need to find \(P\left(B_{2} \cap A\right)\).
c). We need to find \(P\left(B_{1} / A\right)\).
d). We need to find \(P\left(B_{2} / A\right)\).
a).
Now we have to find \(P\left(B_{1} \cap A\right)\).
The formula for the \(P\left(B_{1} \cap A\right)\) is \(P\left(B_{1} \cap A\right)=P\left(A / B_{1}\right) P\left(B_{1}\right)\)
We substitute \(P\left(A / B_{1}\right)\) and \(P\left(B_{1}\right)) values.
\(\begin{array}{l}
P\left(B_{1} \cap A\right)=0.30 \times 0.75 \\
P\left(B_{1} \cap A\right)=0.225
\end{array}\)
Therefore, \(P\left(B_{1} \cap A\right)=0.225\)