Q 3.14 - A First Course in Probability

Question

 An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that (a) the first 2 balls selected are black and the next 2 are white; (b) of the first 4 balls selected, exactly 2 are black. 

Step-by-Step Solution

Answer

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Step 1 Given Information

Given that an urn initially contains 5 white and 7 black balls.

 Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color 

We have to find the probability that  the first 2 balls selected are black and the next 2 are white;

Also find the probability of the first 4 balls selected, exactly 2 are black. 

 

Step 2 Explanation -1

Given an urn contains 12 balls.

Urn: White 5, Black 7

It is also given that for each selection of a ball after noting the color of the ball is replaced along with 2 other balls of the same color are replaced.

Step 3 Explanation-(a)

We have to find the probability that the first two are black and the next two are black. Selection of two black balls:

The probability of selecting 1 black ball from 7 black balls from the urn having 12 balls is . Here, after selecting a black ball, replace the black with another 2 black balls. so the urn has.$\frac{7}{12}$

Here, after selecting a black ball, replace the black with another 2 black balls, so the urn has, Urn: White 5, Black 9

Probability of getting second black ball from 9

black balls from the urn having 14 balls is $\frac{9}{14}$

Here, selected a black ball, so replace it in the

urn along with another 2 black balls, so the urn

has,

Urn: White 5, Black 11

Step 4 Explanation

The following possibilities.

$\left\{\right(BBWW\left)\right(WBBW\left)\right(WWBB\left)\right(BWWB\left)\right(BWBW\left)\right(WBWB\left)\right\}$

Probabilities are equally likely, so to obtain the required probability, add all four probabilities.