| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Probability Generating Functions |
| Type | Derive standard distribution PGF |
| Difficulty | Challenging +1.2 This is a structured PGF question with clear scaffolding through three parts. Part (i) is trivial algebraic verification, part (ii) requires standard PGF derivation for a discrete uniform distribution (textbook material), and part (iii) involves coefficient extraction using binomial expansion. While it requires multiple techniques and careful algebra, the question provides significant guidance and uses well-established methods without requiring novel insight. Slightly above average due to the algebraic manipulation required in part (iii). |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.01a Permutations and combinations: evaluate probabilities |
6 (i) Verify that $\left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + t ^ { 3 } + t ^ { 4 } + t ^ { 5 } \right)$.\\
(ii) An unbiased six-faced die is rolled $r$ times. Show that the probability generating function for the total score is
$$\left[ \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) } \right] ^ { r }$$
(iii) Hence show that the probability of the total score being ( $r + 3$ ) is
$$\left( \frac { 1 } { 6 } \right) ^ { r + 1 } r ( r + 1 ) ( r + 2 )$$
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q6}}