Q. 2.27 - A First Course in Probability

Question

An urn contains $3$red and $7$black balls. Players$A$$\&$ $B$withdraw balls from the urn consecutively until a red ball is selected. Find the probability that $A$selects the red

ball. ($A$draws the first ball, then$B$, and so on. There is no replacement of the balls drawn.)

Step-by-Step Solution

Answer

The probability that A selects the red ball is$0.5833$.

Step 1 Given Information.

An urn contains$3$ red and$7$ black balls. Players$A$$\&$ B withdraw balls from the urn consecutively until a red ball is selected.

Step 2 Explanation.

$A$wins if the first red ball is drawn $1$st, $3$rd, $5$th, or $7$th.

$E\left(1\right)=\left(9C2\right)E\left(3\right)=\left(7C2\right)E\left(5\right)=\left(5C2\right)E\left(7\right)=\left(3C2\right)$

$S=(10 C 3)$

$$\begin{gathered} P\left(Awins\right)=\left[\right(9 C 2)+(7 C 2)+(5 C 2)+(3 C 2\left)\right] /(10 C 3) \\ P\left(Awins\right)=0.5833 \end{gathered}$$