1. Probability Questions

A bag contains 4 Red and 3 Green balls. Helen and Tony play a game where, starting with Helen, they alternately draw a ball at random and do not replace it. The game …finishes when a Red ball is drawn.

It is decided that each player will receive, from the other, n units if they draw the Red ball on the nth draw. Construct a table for the random variable X representing Helen’s winnings: thus X = 1 if Helen draws a Red on the fi…rst draw, -2 if Tony draws it on the next, etc.
Calculate E(X). How much, approximately, would Helen win in a series of 100 games?

2. Hello, mr_motivator!

A bag contains 4 Red and 3 Green balls.
Helen and Tony play a game where, starting with Helen,
they alternately draw a ball at random and do not replace it.
The game finishes when a Red ball is drawn.

It is decided that each player will receive, from the other $n$ units
if they draw the Red ball on the $n^{th}$ draw.

Construct a table for the random variable $X$ representing Helen’s winnings:
thus $X = 1$ if Helen draws a Red on the first draw,
$\text{-}2$ if Tony draws it on the next, etc.

Calculate $E(X)$. How much, approximately, would Helen win in a series of 100 games?
The tree diagram looks like this . . .

. . . $\begin{array}{ccccc}
& * \\
& \frac{4}{7}\swarrow\searrow\frac{3}{7} \\
\end{array}$

Helen's winnings look like this:

. . $\begin{array}{|c|ccc|}\hline
X & \text{Prob} & &\\ \hline \hline
+1 & \frac{4}{7} & = & \frac{4}{7}\\ \\[-4mm]
-2 & \frac{3}{7}\cdot\frac{4}{6} &=& \frac{2}{7} \\ \\[-4mm]
+3 & \frac{3}{7}\cdot\frac{2}{6}\cdot\frac{4}{5} &=& \frac{4}{35} \\ \\[-4mm]
-4 & \frac{3}{7}\cdot\frac{2}{6}\cdot\frac{1}{5}\cdot\f rac{4}{4} &=&\frac{1}{35}\\ \end{array}$

$E(X) \;=\;\tfrac{4}{7}(+1) + \tfrac{2}{7}(-2) + \tfrac{4}{35}(+3) + \tfrac{1}{35}(-4) \;=\;\frac{8}{35}$

She can expect to win an average of $\tfrac{8}{35}$ units per game.

In 100 games, she would win about: . $100\cdot\tfrac{8}{35} \:=\:22\tfrac{6}{7}$ units.