Chapter 3: Q. 3.4 (page 109)

Urn $A$ contains $2$ white balls and $1$ black ball, whereas urn $B$ contains $1$ white ball and $5$ black balls. A ball is drawn at random from urn $A$ and placed in urn $B$ . A ball is then drawn from urn $B$ . It happens to be white. What is the probability that the ball transferred was white?

Short Answer

Expert verified

By Bayer's formula,the colour was trasfered to the ball at $P (W_{A} ∣ W_{B}) = \frac{4}{5} .$

Step by step solution

Step: 1 Events:

The white ball drawn from Urn $A$ - $W_{A}$

The white ball drawn from Urn $B$ - $W_{B}$

The number of balls from $A$ is

$P (W_{A}) = \frac{2}{3} P (W_{A}^{c}) = \frac{1}{3}$

The ball transfers from $A$ to $B$ ,the conditional distribution of ball $B$ is

$\begin{aligned} P (W_{B} ∣ W_{A}) = \frac{2}{7} \\ P (W_{B} ∣ W_{A}^{c}) = \frac{1}{7} \end{aligned}$

Step: 2 Finding values:

By Bayes formula,

$P (W_{A} ∣ W_{B}) = \frac{P (W_{B} ∣ W_{A}) P (W_{A})}{P (W_{B} ∣ W_{A}) P (W_{A}) + P (W_{B} ∣ W_{A}^{c}) P (W_{A}^{c})} P (W_{A} ∣ W_{B}) = \frac{(\frac{2}{7} \times \frac{2}{3})}{(\frac{2}{7} \times \frac{2}{3}) + (\frac{1}{7} \times \frac{1}{3})} P (W_{A} ∣ W_{B}) = \frac{4}{5} .$

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Urn $A$ contains $2$ white balls and $1$ black ball, whereas urn $B$ contains $1$ white ball and $5$ black balls. A ball is drawn at random from urn $A$ and placed in urn $B$ . A ball is then drawn from urn $B$ . It happens to be white. What is the probability that the ball transferred was white?

Short Answer

Step by step solution

Step: 1 Events:

Step: 2 Finding values:

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Urn Acontains 2white balls and 1black ball, whereas urn Bcontains1white ball and 5black balls. A ball is drawn at random from urn Aand placed in urn B. A ball is then drawn from urn B. It happens to be white. What is the probability that the ball transferred was white?

Short Answer

Step by step solution

Step: 1 Events:

Step: 2 Finding values:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Urn $A$ contains $2$ white balls and $1$ black ball, whereas urn $B$ contains $1$ white ball and $5$ black balls. A ball is drawn at random from urn $A$ and placed in urn $B$ . A ball is then drawn from urn $B$ . It happens to be white. What is the probability that the ball transferred was white?