| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find maximum n for P(X > n) > threshold |
| Difficulty | Moderate -0.3 This is a straightforward application of geometric distribution with clearly defined success probability (p=1/3). Parts (i)-(iii) require direct formula application with minimal calculation, while part (iv) involves solving an inequality using logarithms—a standard technique. The question is slightly easier than average because it's well-structured with no conceptual surprises, though it does require familiarity with geometric distribution properties. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| 6(i) | Mean = 3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(ii) | P(X = 6) = q5 p with p = 1/3, q = 2/3 (AEF) | |
| = 32/729 or 0⋅0439 | B1 | Find probability of score of 3 or 4 on exactly 6 throws |
| Answer | Marks | Guidance |
|---|---|---|
| 6(iii) | P(X > 4) = q4 = 16/81 or 0⋅1975 or 0⋅198 | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(iv) | 1 – qn – 1 < 0⋅95 (AEF) | M1 |
| 0⋅05 < (2/3) n – 1 , n – 1 < log 0⋅05 / log 2/3 | M1 | Set q = 2/3, rearrange and take logs (any base) to give bound |
| Answer | Marks | Guidance |
|---|---|---|
| max | A1 | Find n (> or = can earn M1 M1 A0, max 2/3) |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
--- 6(i) ---
6(i) | Mean = 3 | B1 | State mean of X
1
--- 6(ii) ---
6(ii) | P(X = 6) = q5 p with p = 1/3, q = 2/3 (AEF)
= 32/729 or 0⋅0439 | B1 | Find probability of score of 3 or 4 on exactly 6 throws
1
--- 6(iii) ---
6(iii) | P(X > 4) = q4 = 16/81 or 0⋅1975 or 0⋅198 | M1 A1 | Find probability of score of 3 or 4 on more than 4 throws
2
--- 6(iv) ---
6(iv) | 1 – qn – 1 < 0⋅95 (AEF) | M1 | Formulate condition for n (1 – qn is M0)
0⋅05 < (2/3) n – 1 , n – 1 < log 0⋅05 / log 2/3 | M1 | Set q = 2/3, rearrange and take logs (any base) to give bound
n – 1 < 7⋅39, n = 8
max | A1 | Find n (> or = can earn M1 M1 A0, max 2/3)
max
3
Question | Answer | Marks | GuidanceA fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable $X$.
(i) State the mean value of $X$. [1]
(ii) Find the probability that obtaining a 3 or a 4 takes exactly 6 throws. [1]
(iii) Find the probability that obtaining a 3 or a 4 takes more than 4 throws. [2]
(iv) Find the greatest integer $n$ such that the probability of obtaining a 3 or a 4 in fewer than $n$ throws is less than 0.95. [3]
\hfill \mbox{\textit{CAIE FP2 2019 Q6 [7]}}