An urn contains two white and two black balls. We draw balls from the urn randomly and stop after we find a black ball. What is the expectation of the total number of the drawn balls? ◦ $1\frac{1}{2}$ ◦ $1\frac{2}{3}$ ◦ $2$ ◦ $2\frac{1}{3}$ ◦ $2\frac{1}{2}$
My solution: Possible outcomes a. WWB b. WB c. B
$P(outcome a) = \frac{2}{4} * \frac{2}{3} * \frac{2}{2} = \frac{1}{3}$ $P(outcome b) = \frac{2}{4} * \frac{2}{3} * \frac{2}{2} = \frac{1}{3}$ $P(outcome c) = \frac{2}{4} = \frac{1}{2}$
Expectation = $\frac{1}{3} * \frac{1}{3} * \frac{1}{2} * 2 = 2\frac{1}{3}$
probability we are done in first draw =$\frac{1}{2}$
probability we are done in 2nd draw=$\frac{1\times2}{2\times3}=\frac{1}{3}$
probability we are done at 3rd draw=$\frac{1}{6}$
Expectation =$\frac{1}{2}\times1+\frac{1}{3}\times2+\frac{1}{6}\times3$=$\frac{5}{3}$