A2 June 2019 Q2
2. Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed.
One morning Indre took her break time in 4 periods of 5 minutes each.
On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
| Scheme | Marks | AO |
|---|---|---|
| {Let \(C\) = no of calls in a 20 min period} \(C \sim \mathrm{Po}(\ldots)\) | M1 | 3.3 |
| 80 calls per 4-hour period gives \(\dfrac{20}{3}\) per 20 mins i.e. \(C \sim \mathrm{Po}\left(\dfrac{20}{3}\right)\) \([\mathrm{P}(C \gt 4)] = 1 - \mathrm{P}(C \leqslant 4)\) | M1 | 3.4 |
| \(= 0.79437\ldots\) awrt 0.794 | A1 | 1.1b |
| (3) |
Notes
1st M1 for selecting a Poisson model – written or used. May be implied by 2nd M1 or a correct Answer.
2nd M1 for the correct Poisson \(\mathrm{Po}\left(\frac{20}{3}\right)\) or \(\mathrm{Po}(6.67)\) or better seen and writing or using \(1 - \mathrm{P}(C \leqslant 4)\)
A1 for awrt 0.794 (correct ans with no incorrect working scores 3/3)
| Scheme | Marks | AO |
|---|---|---|
| {\(X\) = no. of 5 min periods with no calls} \(X \sim \mathrm{B}\left(4, \mathrm{e}^{-\frac{5}{3}}\right)\) | M1 | 3.3 |
| \(\mathrm{P}(X = 3) = 0.02186125\ldots\) awrt 0.0219 | A1 | 1.1b |
| (2) |
Notes
M1 for selecting a correct model \(\mathrm{B}(4, 0.189)\) or better (calc: \(0.188875\ldots\))
A1 for using the model to get awrt 0.0219 (correct ans with no incorrect working scores 2/2)
| Scheme | Marks | AO |
|---|---|---|
| P(exactly one call) \(\mathrm{e}^{-\frac{5}{3}} \times \dfrac{5}{3}\) or \(\mathrm{e}^{-5} \times 5\) | M1 | 2.1 |
| P(exactly one call in each break) \(= \left(\mathrm{e}^{-\frac{5}{3}} \times \dfrac{5}{3}\right) \times \left(\mathrm{e}^{-5} \times 5\right)\) | M1 | 1.1b |
| \(= 0.0106052\ldots\) awrt 0.0106 | A1 | 1.1b |
| (3) | ||
| (8 marks) |
Notes
1st M1 for a correct prob of 1 call (expressions in e or values)
(allow \(0.31479\ldots\) or awrt 0.315 or \(0.033689\ldots\) or awrt 0.0337)
2nd M1 for a correct probability statement or expression.
E.g. \(\mathrm{P}\left(S = 1 \mid S \sim \mathrm{Po}\left(\frac{5}{3}\right)\right) \times \mathrm{P}(T = 1 \mid T \sim \mathrm{Po}(5))\)
SC e.g. \(F \sim \mathrm{Po}(\lambda)\) used in (b) to find \(\mathrm{P}(F = 0)\)
Then if we see \(Y \sim \mathrm{Po}(3\lambda)\) and statement \(\mathrm{P}(F = 1) \times \mathrm{P}(Y = 1)\) award M0M1
A1 for awrt 0.0106 (correct ans with no incorrect working scores 3/3)