An urn contains one red chip and one white chip.
One chip is drawn at random. If the chip selected is red,
that chip together with two additional red chips are put
back into the urn. If a white chip is drawn, the chip is
returned to the urn. Then a second chip is drawn. What
is the probability that both selections are red?
The answer I keep obtaining is 2/3 but the actual textbook answer is 3/8.
Best Answer
In order to draw two red chips, you must draw the red chip the first time; this happens with probability $\frac12$. If you do draw the red chip initially, the second draw is from an urn with three red chips and one white chip, so the probability of drawing a second red chip is $\frac34$. Thus, the probability of drawing two red chips is $\frac12\cdot\frac34=\frac38$.
Note that a quick sanity check can tell you that $\frac23$ is impossible: the probability cannot be greater than the probability of drawing a red chip the first time, which is clearly $\frac12$.