June 2024 Paper 2 Q12
12 Ryan has to choose one student at random from a group of 11 students. Ryan makes the choice using a single throw of two fair, six-sided dice, together with the following table.
| Total score on the two dice | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Student chosen | A | B | C | D | E | F | G | H | I | J | K |
Sasha suggests making the choice using a single throw of two fair, six-sided dice, together with the following table.
| Scores on the two dice | 1, 1 | 1, 2 | 2, 1 | 1, 3 | 3, 1 | 1, 4 | 4, 1 | 1, 5 | 5, 1 | 1, 6 | 6, 1 |
| Student chosen | A | B | C | D | E | F | G | H | I | J | K |
Ryan says that a further instruction is needed to complete the method.
| Scheme | Marks | AO |
|---|---|---|
| Two correct, unequal probabilities: e.g. \(\mathrm{P}(2) = \frac{1}{36},\ \mathrm{P}(3) = \frac{2}{36}\) | M1 | 2.4 |
| Probabilities not equal (and hence not random). | A1 | 3.5b |
| [2] |
Notes
M1: Must be seen. Condone a correct statement e.g. “The probability of obtaining 2 is not the same as the probability of getting 3” – must refer to specific scores, need not compute probabilities (but if given these must be correct).
A1: A generalised conclusion must be present:
- Allow “probabilities are not the same”
- May come at the beginning e.g. “the chance of each student being chosen is not equal because…”
Alternative using combinations
| Scheme | Marks |
|---|---|
| Demonstrate that there are more ways of obtaining one answer than another e.g. “there are more ways of obtaining 4 than 2” | M1 |
| (Hence student C is more likely to be chosen than student A) and therefore the probabilities are not equal (and hence not random). | A1 |
M1: Must be seen. Must refer to specific scores but need not compute the number of combinations/ways (but if given these must be correct).
A1: A generalised conclusion must be present:
- Accept “student __ is more likely to be chosen than student __, therefore it is not random”
- Accept “some scores are more likely than others”
| Scheme | Marks | AO |
|---|---|---|
| (i) Throw repeatedly until one of these pairs is obtained | B1 | 3.5c |
| [1] | ||
| (ii) \(\dfrac{1}{11}\) | B1 | 3.4 |
| [1] |
Notes
(b)(i) B1: Any valid correction to the method, e.g.:
- “If obtain a pair not included, throw again.”
- “make the two dice distinguishable” (because 1,3 and 3,1 have different students) or “throw in order”
Condone extending to the entire sample space (e.g. “assign a further 2 pairs to each student and the remaining 3 to ‘throw again’”) but the method must remain random.
(b)(ii) B1: Accept \(\frac{1}{36}\)