jmc

Question

There are four red balls and two white balls in a jar. One ball is randomly removed and replaced with a ball of the opposite color. The jar is then shaken and one ball is randomly selected. What is the probability that this ball is red? Express your answer as a common fraction.

Accepted Answer

We split the problem into two cases.

Case I: A red ball is removed. The probability that a red ball is removed is

\frac{4}{6} = \frac{2}{3}

. After it is replaced by a white ball, the probability of drawing a red ball is

\frac{1}{2}

. Thus, the probability that a red ball will be drawn in this case is

\frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3}

.

Case II: A white ball is removed. The probability that a white ball is removed is

\frac{2}{6} = \frac{1}{3}

. After it is replaced by a red ball, the probability of drawing a red ball is

\frac{5}{6}

. Thus, the probability that a red ball will be drawn in this case is

\frac{5}{18}

.

We add the two probabilities for a total probability of

\frac{11}{18}

.