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.

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.

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.

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

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.