| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2021 |
| Session | January |
| Marks | 12 |
| Topic | Geometric Distribution |
| Type | Variance of geometric distribution |
| Difficulty | Standard +0.3 This is a straightforward geometric distribution question requiring standard formulas and basic expectation calculations. Part (a) uses direct probability formulas, part (b) applies the variance formula to find E(B²), and part (c) compares expected values using given results. While it involves multiple parts and requires careful application of formulas, it's a textbook exercise with no novel problem-solving or proof required, making it slightly easier than average. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
A spinner can land on red or blue. When the spinner is spun, there is a probability of $\frac{1}{3}$ that it lands on blue. The spinner is spun repeatedly.
The random variable $B$ represents the number of the spin when the spinner first lands on blue.
\begin{enumerate}[label=(\alph*)]
\item Find \begin{enumerate}[label=(\roman*)]
\item P($B = 4$)
\item P($B \leq 5$)
\end{enumerate}
[4]
\item Find E($B^2$) [3]
\end{enumerate}
Steve invites Tamara to play a game with this spinner.
Tamara must choose a colour, either red or blue.
Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable $X$ represents the number of the spin when this occurs.
If Tamara chooses red, her score is $e^X$
If Tamara chooses blue, her score is $X^2$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q6 [12]}}