Higher November 2022 Paper 3 Q20
20 There should be a train leaving a station every hour from 7 am
No trains leave early.
P(the first train leaves on time) = 0.9
For all the other trains,
if the previous train did leave on time, P(this train leaves on time) = 0.8
if the previous train did not leave on time, P(this train leaves on time) = 0.65
(a) Work out \(\quad\) P(the first three trains leave on time) [2 marks]
(b) The 2 pm train does not leave on time.
Work out \(\quad\) P(exactly one of the next two trains does not leave on time) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(0.9 \times 0.8^2\) or \(0.9 \times 0.64\) | M1 | oe |
| 0.576 or 0.58 or \(\dfrac{72}{125}\) | A1 | oe fraction decimal or percentage |
Additional guidance
| Ignore any attempt to convert a correct answer | M1A1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| (late, on time =) \((1 - 0.65) \times 0.65\) or \(0.35 \times 0.65\) or 0.2275 or (on time, late =) \(0.65 \times (1 - 0.8)\) or \(0.65 \times 0.2\) or 0.13 | M1 | may be seen on tree diagram |
| \((1 - 0.65) \times 0.65 + 0.65 \times (1 - 0.8)\) or \(0.2275 + 0.13\) | M1dep | oe |
| 0.3575 or \(\dfrac{143}{400}\) | A1 | oe fraction, decimal or percentage Accept 0.358 or 0.36 with M1 scored |
| Alternative method 2 | ||
| (late, late =) \((1 - 0.65)^2\) or \(0.35^2\) or 0.1225 or (on time, on time =) \(0.65 \times 0.8\) or 0.52 | M1 | may be seen on tree diagram |
| \(1 - (1 - 0.65)^2 - 0.65 \times 0.8\) or \(1 - 0.1225 - 0.52\) | M1dep | oe |
| 0.3575 or \(\dfrac{143}{400}\) | A1 | oe fraction, decimal or percentage Accept 0.358 or 0.36 with M1 scored |
Additional guidance
| Up to M2 may be awarded for correct work, with no or incorrect answer, even if this is seen amongst multiple attempts | |
| Ignore any attempt to convert a correct answer | M1M1A1 |