S2 June 2007 Q5
5.
(a) Write down the conditions under which the Poisson distribution may be used as an approximation to the Binomial distribution. (2)
A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of the caller being connected to the wrong agent is 0.01
(b) Find the probability that 2 consecutive calls will be connected to the wrong agent. (2)
(c) Find the probability that more than 1 call in 5 consecutive calls are connected to the wrong agent. (3)
The call centre receives 1000 calls each day.
(d) Find the mean and variance of the number of wrongly connected calls. (3)
(e) Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent. (2)
| Scheme | Marks |
|---|---|
| If \(X \sim \mathrm{B}(n, p)\) and | |
| \(n\) is large, \(n \gt 50\) | B1 |
| \(p\) is small, \(p \lt 0.2\) | B1 |
| then \(X\) can be approximated by \(\mathrm{Po}(np)\) | |
| (2) |
| Scheme | Marks |
|---|---|
| P(2 consecutive calls) \(= 0.01^2\) | M1 |
| \(= 0.0001\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(5, 0.01)\) | B1 |
| \(\mathrm{P}(X \gt 1) = 1 - \mathrm{P}(X = 1) - \mathrm{P}(X = 0)\) \(\qquad\qquad = 1 - 5(0.01)(0.99)^4 - (0.99)^5\) \(\qquad\qquad = 1 - 0.0480298\ldots - 0.95099\ldots\) | M1 |
| \(= 0.00098\) | A1 |
| (3) |
Notes
B1 may be implied
A1 awrt 0.00098
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(1000, 0.01)\) | B1 |
| Mean \(= np = 10\) | B1 |
| Variance \(= np(1 - p) = 9.9\) | B1 |
| (3) |
Notes
1st B1 may be implied by correct mean and variance
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{Po}(10)\) | |
| \(\mathrm{P}(X \gt 6) = 1 - \mathrm{P}(X \leqslant 6)\) \(\qquad\qquad = 1 - 0.1301\) | M1 |
| \(= 0.8699\) | A1 |
| (2) | |
| (12 marks) |
Notes
A1 awrt 0.870