S2 June 2007 Q3
3. An engineering company manufactures an electronic component. At the end of the manufacturing process, each component is checked to see if it is faulty.
Faulty components are detected at a rate of 1.5 per hour.
(a) Suggest a suitable model for the number of faulty components detected per hour. (1)
(b) Describe, in the context of this question, two assumptions you have made in part (a) for this model to be suitable. (2)
(c) Find the probability of 2 faulty components being detected in a 1 hour period. (2)
(d) Find the probability of at least one faulty component being detected in a 3 hour period. (3)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{Po}(1.5)\) | B1 |
| (1) |
Notes
B1 need Po and 1.5
| Scheme | Marks |
|---|---|
| Faulty components occur at a constant rate. Faulty components occur independently or randomly. Faulty components occur singly. | B1 B1 |
| (2) |
Notes
any two of the 3 only need faulty once
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 2) = \mathrm{P}(X \leqslant 2) - \mathrm{P}(X \leqslant 1)\) or \(\dfrac{\mathrm{e}^{-1.5}(1.5)^2}{2}\) | M1 |
| \(= 0.8088 - 0.5578\) \(= 0.251\) | A1 |
| (2) |
Notes
A1 awrt 0.251
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{Po}(4.5)\) | B1 |
| \(\mathrm{P}(X \geqslant 1) = 1 - \mathrm{P}(X = 0)\) \(\qquad\qquad\ = 1 - \mathrm{e}^{-4.5}\) \(\qquad\qquad\ = 1 - 0.0111\) | M1 |
| \(= 0.9889\) | A1 |
| (3) | |
| (8 marks) |
Notes
B1 4.5 may be implied
A1 awrt 0.989