Question

Suppose the events ${\mathbf{B}}_1$and ${\mathbf{B}}_2$are mutually exclusive and complementary events, such that$\mathbf{P}\mathbf{(}{\mathbf{B}}_1\mathbf{)}\mathbf{=}\mathbf{.}\mathbf{75}$and$\mathbf{P}\mathbf{(}{\mathbf{B}}_2\mathbf{)}\mathbf{=}\mathbf{.}\mathbf{25}$ Consider another event A such that role="math" localid="1658212959871" $\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{1}}}\mathbf{\right)}\mathbf{=}\mathbf{.}\mathbf{3}$, role="math" localid="1658213029408" $\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{2}}}\mathbf{\right)}\mathbf{=}\mathbf{.}\mathbf{5}$.

1. Find$\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{1}}\mathbf{\cap }\mathbf{A}\mathbf{)}$.
2. Find$\mathbf{P}\mathbf{(}{\mathbf{B}}_2\cap \mathbf{A}\mathbf{)}$
3. Find P(A) using part a and b.
4. Findrole="math" localid="1658213127512" $\mathbf{P}\mathbf{\left(}\frac{{\mathbf{B}}_{\mathbf{1}}}{\mathbf{A}}\mathbf{\right)}$.
5. Findrole="math" localid="1658213164846" $\mathbf{P}\mathbf{\left(}\frac{{\mathbf{B}}_{\mathbf{2}}}{\mathbf{A}}\mathbf{\right)}$.

Short answer

The values of probabilities in all parts are:

1. 0.225
2. 0.125
3. 0.35
4. 0.64
5. 0.357

Step-by-step solution

Important formula

The general rule of multiplication is $\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{)}\mathbf{\times }\mathbf{P}\mathbf{\left(}\frac{\mathbf{B}}{\mathbf{A}}\mathbf{\right)}$.

Find P(B1∩A).

$\begin{array}{l}\mathbf{P}\mathbf{(}{\mathbf{B}}_1\cap \mathbf{A}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{1}}}\mathbf{\right)}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{B}}_1\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{\left)}\mathbf{\right(}\mathbf{0}\mathbf{.}\mathbf{75}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{225}\end{array}$

So, the values of probabilities in all parts are 0.225

Find P(B2∩A)

$\begin{array}{c}\mathbf{P}\mathbf{(}{\mathbf{B}}_2\mathbf{\cap }\mathbf{A}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{2}}}\mathbf{\right)}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{2}}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.25}\mathbf{\left)}\mathbf{\right(}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{=}\mathbf{0.125}\end{array}$

Hence, the values of probabilities in all parts are 0.125

Find P(A)

$\begin{array}{l}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{(}{\mathbf{B}}_1\cap \mathbf{A}\mathbf{)}+\mathbf{P}\mathbf{(}{\mathbf{B}}_2\cap \mathbf{A}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{225}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{125}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}\end{array}$

Accordingly, the values of probabilities in all parts are 0.35

Find P(B1A)

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\frac{{\mathbf{B}}_{\mathbf{1}}}{\mathbf{A}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{1}}\mathbf{\cap }\mathbf{A}\mathbf{)}}{P\left(A\right)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{0}\mathbf{.225}}{\mathbf{0}\mathbf{.35}}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{64}\end{array}$

Henceforth, the values of probabilities in all parts are 0.64

Find P(B2A)

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\frac{{\mathbf{B}}_{\mathbf{2}}}{\mathbf{A}}\mathbf{\right)}=\frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_2\cap \mathbf{A}\mathbf{)}}{P\left(A\right)}\\ =\frac{0.125}{0.35}\\ =0.357\end{array}$

Therefore, the values of probabilities in all parts are 0.357