Geometric & Negative Binomial Distributions (Edexcel A Level Further Maths: Further Statistics 1): Exam Questions

The random variable $X$ has probability generating function ${\text{G}}_X\left(t\right)$, where

${\text{G}}_X\left(t\right)=(9-8t{)}^{-\frac{1}{2}}$

The independent random variables $X_1$, $X_2$, $X_3$ and $X_4$ each have the same distribution as $X$. The random variable $Y$ is defined by

$Y=X_1+X_2+X_3+X_4+2$

By finding the probability generating function of $Y$, state the name of the distribution of $Y$.

Hence, or otherwise, find $\text{P}(X_1+X_2+X_3+X_4=2)$.

The probability of winning a prize when playing a single game of Ringo is $\frac{1}{3}$. When more than one game is played, the games are independent.

Sam plays 12 games.

Find the probability that Sam wins 5 or more prizes.

Tessa plays a series of games.

Find the probability that Tessa wins her 3rd prize on her 10th game.

Rama invites Sam and Tessa to play some new games of Ringo. They must pay Rama £2 for each game they play, but Rama will pay them £3 the first time they win a prize, £6 the second time, and £$\left(3w\right)$ when they win their $w$th prize.

Sam decides to play $n$ games of Ringo with Rama.

Show that Sam's expected profit is $\pounds \frac{1}{6}\left(n^2-7n\right)$.

Given that $n=6$, find the probability that Sam does not make a loss.

Tessa agrees to play Ringo with Rama. She will play games until she wins $r$ prizes and then she will stop.

Find, in terms of $r$, Tessa's expected profit.