| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Topic | Geometric Distribution |
| Type | Geometric with multiple success milestones |
| Difficulty | Standard +0.3 Part (i) is a standard proof about geometric distribution requiring simple algebraic manipulation of the geometric series. Parts (ii) and (iii) involve routine application of formulas: solving q³ = 0.216 gives q = 0.6, then p = 0.4, followed by direct substitution into standard geometric distribution formulas. This is slightly easier than average as it's mostly recall and straightforward calculation with minimal problem-solving. |
| 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 |
**Question 5(i)**
**Alternative version 1:**
- $P(X > n) = 1 - P(X \leqslant n)$
- $= 1 - \{p + pq + \ldots + pq^{n-1}\}$ **M1** ($1 -$ list of first $n$ probabilities)
- $= 1 - \dfrac{p(1-q^n)}{1-q}$ **M1** (Sum of GP used correctly.)
- $= 1 - (1 - q^n) = q^n$ **A1** (Simplified convincingly.)
**Alternative version 2:**
- $P(X > n) = pq^n + pq^{n+1} + pq^{n+2} + \ldots$ **M1** (List of subsequent probabilities.)
- $= q^n\{p + pq + pq^2 + \ldots\}$ **M1** (Sum of infinite GP used.)
- $= q^n \times 1 = q^n$ **A1** (Sum in $\{\} = 1$ (property of Geo($p$)).)
**Alternative version 3:**
- If $X > n$ then $\ldots$ **M1**
- $\ldots$ must "fail" on first $n$ attempts. **M1**
- $\therefore P(X > n) = P(\text{"Fail"}\; n \text{ times}) = q^n$ **A1**
**Question 5(ii)**
- $P(X \geqslant 4) = P(X > 3) = q^3 = 0.216$ **M1** (Use $q^3$ and find $q$.)
- $\therefore q = 0.6$
- $\therefore p = 0.4$ **A1** (cao)
- $P(X \leqslant 8) = 1 - P(X > 8) = 1 - 0.6^8$ **M1** ($1 - q^8$)
- $= 1 - 0.01679616 = 0.98320384 \approx 0.983$ **A1** (FT $1 -$ *their* $q^8$.)
**Question 5(iii)**
- $E(X) = \dfrac{1}{0.4} = 2.5$ **B1** (FT *their* $p$.)
- $\text{Var}(X) = \dfrac{0.6}{0.4^2} = 3.75$ **B1** (FT *their* $p$.)
**Total: 9 marks**5 The random variable $X$ has a geometric distribution: $X \sim \operatorname { Geo } ( p )$.\\
(i) Show that $\mathrm { P } ( X > n ) = q ^ { n }$, where $q = 1 - p$.
You are given that $\mathrm { P } ( X \geqslant 4 ) = 0.216$.\\
(ii) Use the result given in part (i) to find the value of $p$ and $\mathrm { P } ( X \leqslant 8 )$.\\
(iii) Write down $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2017 Q5 [9]}}