Q. 6.37 - A First Course in Probability

Question

In Problem $6.5$, calculate the conditional probability mass function of Y1 given that

(a) $Y_2=1$

(b) $Y_2=0$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) P\left(Y_1=0\mid Y_2=1\right)=0.8464 \\ P\left(Y_1=1\mid Y_2=1\right)=0.1536 \end{gathered}$$

$$\begin{gathered} \left(b\right) P\left(Y_1=0\mid Y_2=0\right)=0.7702 \\ P\left(Y_1=1\mid Y_2=0\right)=0.2297 \end{gathered}$$

Step 1: Given information (part a)

An urn contains $5$ white balls, $8$ red balls, and $3$ balls are to be chosen with replacement. the white balls have been numbered.

Compute the conditional probability that $Y_1$ given that $Y_2=0$

$P\left(Y_1\mid Y_2=1\right)=\frac{P\left(Y_1\cap Y_2=1\right)}{P\left(Y_2=1\right)}$

Step 2: Explanation (part a)

Substitute the values from the joint probability mass function.

For $Y_1=0$

$$\begin{gathered} P\left(Y_1=0\mid Y_2=1\right)=\frac{P\left(Y_1=0\cap Y_2=1\right)}{P\left(Y_2=1\right)} \\ =\frac{0.1807}{0.2135} \\ =0.8464 \\ For Y_1=1 \\ P\left(Y_1=1\mid Y_2=1\right)=\frac{P\left(Y_1=1\cap Y_2=1\right)}{P\left(Y_2=1\right)} \\ =\frac{0.0328}{0.2135} \\ =0.1536 \end{gathered}$$

Step 1: Given information (part b)

Compute the conditional probability that $Y_1$ given that $Y_2=0$

$P\left(Y_1\mid Y_2=0\right)=\frac{P\left(Y_1\cap Y_2=0\right)}{P\left(Y_2=0\right)}$

Step 4: Explanation (part b)

Substitute the values from the joint probability mass function.

$For Y_1=0$

$$\begin{gathered} P\left(Y_1=0\mid Y_2=0\right)=\frac{P\left(Y_1=0\cap Y_2=0\right)}{P\left(Y_2=0\right)} \\ =\frac{0.6058}{0.7865} \\ =0.7702 \\ For Y_1=1 \\ P\left(Y_1=1\mid Y_2=0\right)=\frac{P\left(Y_1=1\cap Y_2=0\right)}{P\left(Y_2=0\right)} \\ =\frac{0.1807}{0.7865} \\ =0.2297 \end{gathered}$$