S2 January 2012 Q5
5. The probability of an electrical component being defective is 0.075
The component is supplied in boxes of 120
(a) Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box. (5)
A retailer buys 2 boxes of components.
(b) Estimate the probability that there are at least 4 defective components in each box. (2)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(120, 0.075)\) | B1 |
| Approximated by Po(9) | M1A1 |
| \(\mathrm{P}(X \gt 3) = 1 - \mathrm{P}(X \leqslant 3)\) | M1 |
| \(= 1 - 0.0212\) \(= 0.9788\) awrt 0.979 | A1 |
| (5) |
Notes
B1 Writing or use of B(120,0.075) may be implied by using Po(9) or N(9,8.325)
1st M1 writing or use of Poisson
1st A1 writing or use of Po(9)
2nd M1 for writing or using \(1 - \mathrm{P}(X \leqslant 3)\) or this may be implied by an awrt 0.972 using normal approximation.
| Scheme | Marks |
|---|---|
| P(At least 4 defective components in each box) \(= \mathrm{P}(X \gt 3) \times \mathrm{P}(X \gt 3)\) | M1 |
| \(= 0.9788^2\) \(= 0.95804944\) awrt 0.958 | A1 |
| (2) | |
| (7 marks) |
Notes
M1 ((their (a))2 or \(0.979^2\) or \(0.9788^2\) or \(0.98^2\)