Higher November 2022 Paper 5 Q12
12 The time, \(t\) seconds, taken by each of 60 students to complete a puzzle is recorded.
The table shows information about these times.
| Time (t seconds) | \(20 \lt t \leqslant 30\) | \(30 \lt t \leqslant 40\) | \(40 \lt t \leqslant 50\) | \(50 \lt t \leqslant 70\) | \(70 \lt t \leqslant 90\) |
|---|---|---|---|---|---|
| Frequency | 8 | 0 | 12 | 30 | 10 |
(a) Two students are picked at random.
Reece works out the probability that they both took longer than 50 seconds to complete the puzzle.
Reece’s working is shown below.
Reece works out the probability that they both took longer than 50 seconds to complete the puzzle.
Reece’s working is shown below.
The number of students who took longer than 50 seconds is 30 + 10 = 40
The probability that one student took longer than 50 seconds is \(\dfrac{40}{60} = \dfrac{2}{3}\)
The probability they both took longer than 50 seconds is \(\dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{4}{9}\)
Explain the error in their method and write the correct calculation that Reece needs to do.
You do not need to work out the answer to the calculation. [2]
(b) Two students are picked at random from those who took 50 seconds or less.
Find the probability that one of them took 30 seconds or less and the other took more than 40 seconds.
You must show your working. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| The two events are dependent oe and \(\frac{40}{60} \times \frac{39}{59}\) isw | 2 | B1 for either | Accept e.g. The second probability is not \(\frac{2}{3}\) oe The second probability is wrong oe The second probability is \(\frac{39}{59}\) There is one less for the second pick oe It is out of 59 for the 2nd pick oe Any incorrect statement is B0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\frac{48}{95}\) oe with correct working | 5 | M4 for \(2\left(\frac{8}{20} \times \frac{12}{19}\right)\) oe or M3 for \(\frac{8}{20} \times \frac{12}{19}\) oe or M2 for \(\frac{8}{20}\) and \(\frac{12}{19}\) or \(\frac{8}{19}\) and \(\frac{12}{20}\) oe seen or M1 for \(\frac{8}{20}\) or \(\frac{12}{20}\) oe seen If 0 or M1 scored, instead award SC2 for \(\frac{8}{n} \times \frac{12}{n - 1}\) oe or for \(2\left(\frac{8}{20} \times \frac{12}{20}\right)\) oe or for answer \(\frac{48}{95}\) oe with no or insufficient working If 0 scored SC1 for \(\frac{8}{n}\) and \(\frac{12}{n - 1}\) or \(\frac{8}{n - 1}\) and \(\frac{12}{n}\) seen or for answers \(\frac{24}{95}\) oe or \(\frac{12}{25}\) oe with no or insufficient working | ‘Correct working’ needs evidence of M2 Must be proper fractions and \(n \leqslant 60\) Must be proper fractions and \(n \leqslant 60\) |