| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find minimum n for P(X ≤ n) > threshold |
| Difficulty | Standard +0.3 This is a straightforward geometric distribution question requiring basic probability calculations. Part (i) involves solving a simple equation (0.8)^4 = 0.4096. Parts (ii) and (iii) apply standard geometric distribution formulas with minimal problem-solving required. The context is clear and the mathematical steps are routine for Further Maths students. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(i) | P(X ⩾ 5) = (1 – p)4 = 0.4096 | |
| p = 1 – 0.8 = 0.2 AG | M1A1 | Verify p using P(X ⩾ 5) = 0.4096 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(ii) | P(X = 6) = (1 – p)5 p = 0.85 × 0.2 = 0.0655 | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(iii) | 1 – (1 – p)N > 0.9 | M1 |
| 0.1 > 0.8N | A1 | (< or = can earn M1 M1 only, max 2/4) |
| N > log 0.1 / log 0.8 = 10.3 | M1 | Rearrange and take logs (any base) to give bound |
| Answer | Marks | Guidance |
|---|---|---|
| min | A1 | Find N and corresponding day and week |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | P(X ⩾ 5) = (1 – p)4 = 0.4096
p = 1 – 0.8 = 0.2 AG | M1A1 | Verify p using P(X ⩾ 5) = 0.4096
2
--- 8(ii) ---
8(ii) | P(X = 6) = (1 – p)5 p = 0.85 × 0.2 = 0.0655 | M1A1 | Find P(X = 6)
2
--- 8(iii) ---
8(iii) | 1 – (1 – p)N > 0.9 | M1 | Formulate condition for N ( 1 – (1 – p)N –1 is M0 )
0.1 > 0.8N | A1 | (< or = can earn M1 M1 only, max 2/4)
N > log 0.1 / log 0.8 = 10.3 | M1 | Rearrange and take logs (any base) to give bound
N = 11 corresponding to Monday of 3rd week
min | A1 | Find N and corresponding day and week
min
4
Question | Answer | Marks | GuidanceLan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value $p$. The random variable $X$ denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096.
\begin{enumerate}[label=(\roman*)]
\item Show that $p = 0.2$.
[2]
\item Find the probability that Lan first gets a seat on Monday of the second week in his new job.
[2]
\item Find the least integer $N$ such that $\text{P}(X \leqslant N) > 0.9$, and identify the day and the week that corresponds to this value of $N$.
[4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2018 Q8 [8]}}