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Statistics / A First Course in Probability 9 / Chapter 3 / Problem 30P

A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross

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Consider two boxes, one containing 1 black and 1 white

Chapter 3, Problem 30P

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QUESTION:

Problem 30P

Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. A box is selected at random, and a marble is drawn from it at random. What is the probability that the marble is black? What is the probability that the first box was the one selected given that the marble is white?

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QUESTION:

Problem 30P

Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. A box is selected at random, and a marble is drawn from it at random. What is the probability that the marble is black? What is the probability that the first box was the one selected given that the marble is white?

ANSWER:

Step 1 of 2

(a)

We are asked to find the probability that the marble is black and also the probability that the first box was the one selected given that the marble is white.

We have given two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble.

Let $B$ be the event that the drawn ball is black and let ${X}_{1}({X}_{2})$ be the event that we select the first ( second) box. Then to calculate $P(B)$ we will condition on the box drawn from as

$P(B)=P(B|{X}_{1})P({X}_{1})+P(B|{X}_{2})P({X}_{2})$

Now $P(B|{X}_{1})=0.5,\ P(B|{X}_{2})=2/3,\ and\ P({X}_{1})=P({X}_{2})=0.5$

$P(B)=0.5\times{}0.5+2/3\times{}0.5=\frac{7}{12}$

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