S2 June 2014 Q1
1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
(a) Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
(i) exactly 7
(ii) at least 10
(5)A patient arrives at 11.30 a.m.
(b) Find the probability that the next patient arrives before 11.45 a.m. (3)
| Scheme | Marks |
|---|---|
| Po(9) | B1 |
| (i) \(\mathrm{P}(X \leqslant 7) - \mathrm{P}(X \leqslant 6) = 0.3239 - 0.2068\) \(\dfrac{\mathrm{e}^{-9}9^7}{7!}\) | M1 |
| \(= 0.1171\) | A1 |
| (ii) \(\mathrm{P}(X \geqslant 10) = 1 - \mathrm{P}(X \leqslant 9)\) | M1 |
| \(= 1 - 0.5874\) \(= 0.4126\) | A1 |
| (5) |
Notes
B1 Po(9) written or used in either (i) or (ii)
(i) M1 writing \(\mathrm{P}(X \leqslant 7) - \mathrm{P}(X \leqslant 6)\) or \(\dfrac{\mathrm{e}^{-\lambda}\lambda^7}{7!}\)
This may be implied by 0.3239 – 0.2068
A1 awrt 0.117
(ii) M1 writing \(1 - \mathrm{P}(X \leqslant 9)\)
This may be implied by 1 – 0.5874.
A1 awrt 0.413
| Scheme | Marks |
|---|---|
| Po(1.5) | B1 |
| P(next patient before 11:45) = 1- P(0) \(= 1 - \mathrm{e}^{-1.5}\) | M1 |
| \(= 0.7769\) | A1 |
| (3) | |
| (8 marks) |
Notes
B1 Po(1.5) written or used
M1 writing or using 1 – P(0) or \(1 - \mathrm{e}^{-\lambda}\).
This may be implied by 1 – 0.2231
A1 awrt 0.777