A bag contains $r$ red balls and $t$ white balls. In each draw, a ball is randomly selected, its color is observed, and then it is returned to the bag along with $a$ more balls of the same color. The process is repeated four times. We need to find the probability that the first and second draws result in red balls, while the third and fourth draws result in white balls.

1. Problem Description

A bag contains $r$ red balls and $t$ white balls. In each draw, a ball is randomly selected, its color is observed, and then it is returned to the bag along with $a$ more balls of the same color. The process is repeated four times. We need to find the probability that the first and second draws result in red balls, while the third and fourth draws result in white balls.

2. Solution Steps

Let $R$ denote the event of drawing a red ball and $W$ denote the event of drawing a white ball. We are looking for the probability $P(RRWW)$.

First draw: The probability of drawing a red ball is $P(R_1) = \frac{r}{r+t}$.

After the first draw, the bag contains $r+a$ red balls and $t$ white balls, for a total of $r+t+a$ balls.

Second draw: The probability of drawing a red ball, given that the first ball was red, is $P(R_2|R_1) = \frac{r+a}{r+t+a}$.

After the second draw, the bag contains $r+2a$ red balls and $t$ white balls, for a total of $r+t+2a$ balls.

Third draw: The probability of drawing a white ball, given that the first two balls were red, is $P(W_3|R_1, R_2) = \frac{t}{r+t+2a}$.

After the third draw, the bag contains $r+2a$ red balls and $t+a$ white balls, for a total of $r+t+3a$ balls.

Fourth draw: The probability of drawing a white ball, given that the first two balls were red and the third was white, is $P(W_4|R_1, R_2, W_3) = \frac{t+a}{r+t+3a}$.

The probability of the sequence RRWW is:

$P(RRWW) = P(R_1) \cdot P(R_2|R_1) \cdot P(W_3|R_1, R_2) \cdot P(W_4|R_1, R_2, W_3)$

$P(RRWW) = \frac{r}{r+t} \cdot \frac{r+a}{r+t+a} \cdot \frac{t}{r+t+2a} \cdot \frac{t+a}{r+t+3a}$

3. Final Answer

The probability is $\frac{r(r+a)t(t+a)}{(r+t)(r+t+a)(r+t+2a)(r+t+3a)}$.