Chapter 6: Q. 6.37 (page 273)

In Problem $6.5$ , calculate the conditional probability mass function of Y1 given that
(a) $Y_{2} = 1$
(b) $Y_{2} = 0$

Short Answer

Expert verified

$(a) P (Y_{1} = 0 ∣ Y_{2} = 1) = 0.8464 P (Y_{1} = 1 ∣ Y_{2} = 1) = 0.1536$

$(b) P (Y_{1} = 0 ∣ Y_{2} = 0) = 0.7702 P (Y_{1} = 1 ∣ Y_{2} = 0) = 0.2297$

Step by step solution

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 $Y_{2} = 0$

role="math" localid="1647264516934" $P (Y_{1} ∣ Y_{2} = 1) = \frac{P (Y_{1} \cap Y_{2} = 1)}{P (Y_{2} = 1)}$

Explanation (part a)

Substitute the values from the joint probability mass function.

For $Y_{1} = 0$

role="math" localid="1647265027423" $P (Y_{1} = 0 ∣ Y_{2} = 1) = \frac{P (Y_{1} = 0 \cap Y_{2} = 1)}{P (Y_{2} = 1)} = \frac{0.1807}{0.2135} = 0.8464 F o r Y_{1} = 1 P (Y_{1} = 1 ∣ Y_{2} = 1) = \frac{P (Y_{1} = 1 \cap Y_{2} = 1)}{P (Y_{2} = 1)} = \frac{0.0328}{0.2135} = 0.1536$

Given information (part b)

Compute the conditional probability that $Y_{1}$ given that $Y_{2} = 0$

$P (Y_{1} ∣ Y_{2} = 0) = \frac{P (Y_{1} \cap Y_{2} = 0)}{P (Y_{2} = 0)}$

Explanation (part b)

Substitute the values from the joint probability mass function.

$F o r Y_{1} = 0$

$P (Y_{1} = 0 ∣ Y_{2} = 0) = \frac{P (Y_{1} = 0 \cap Y_{2} = 0)}{P (Y_{2} = 0)} = \frac{0.6058}{0.7865} = 0.7702 F o r Y_{1} = 1 P (Y_{1} = 1 ∣ Y_{2} = 0) = \frac{P (Y_{1} = 1 \cap Y_{2} = 0)}{P (Y_{2} = 0)} = \frac{0.1807}{0.7865} = 0.2297$

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In Problem $6.5$ , calculate the conditional probability mass function of Y1 given that
(a) $Y_{2} = 1$
(b) $Y_{2} = 0$

Short Answer

Step by step solution

Given information (part a)

Explanation (part a)

Given information (part b)

Explanation (part b)

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In Problem 6.5, calculate the conditional probability mass function of Y1 given that (a) Y2=1(b)Y2=0

Short Answer

Step by step solution

Given information (part a)

Explanation (part a)

Given information (part b)

Explanation (part b)

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

In Problem $6.5$ , calculate the conditional probability mass function of Y1 given that
(a) $Y_{2} = 1$
(b) $Y_{2} = 0$