| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 11 |
| Topic | Geometric Distribution |
| Type | Geometric with multiple success milestones |
| Difficulty | Standard +0.8 This question requires understanding of geometric distribution across multiple contexts: basic probability calculation, conditional probability with memoryless property, solving equations involving geometric probabilities, and conditional expectation. Part (iii) requires algebraic manipulation to solve for n, and part (iv) demands careful reasoning about conditional expectations given a specific first success, going beyond standard textbook exercises. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
(i) Modelling assumption: whether it rains or not on any particular day is independent of whether it has rained or been dry previously [B1]
The geometric distribution Geo(0.25) is to be used [M1]
Calculate $0.75^6 \times 0.25 = 0.0445$ [A1] **3 marks**
(ii) Calculate $0.75^3 \times 0.25 = 0.105$ [B1] **1 mark**
(iii) Using Geo(0.25) for two successive independent periods [M1]
$(0.75^{n-1} \times 0.25) \times (0.75^2 \times 0.25) = 0.02$ [A1]
Use of logs or trial and improvement to solve $0.75^{n+1} = 0.32$ [M1]
Obtain $n = 3$ [A1] **4 marks**
(iv) Use of 'expected number of wet days in any period of $N$ days is $0.25N$' or equivalent for dry days [M1]
Expected number of wet days is $1 + 22 \times 0.25$ [M1]
Expected number of dry days is 23.5 [A1] **3 marks**11 During the 30-day month of April, the probability that it will rain on any given day is 0.25 .\\
(i) Find the probability that the first rainy day in April is the 7th of April, explaining any modelling assumptions you have made.\\
(ii) Given that it does not rain on the first 6 days of April, find the probability that it first rains on the 10th of April.\\
(iii) The probability that it first rains on the $n$th of April and next on the ( $n + 3$ )th of April is 0.02 , correct to 1 significant figure. Determine $n$.\\
(iv) Determine the expected number of dry days in April, given that it first rains on the 8th of April.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q11 [11]}}