Higher November 2021 Paper 2 Q22
22 Liam is trying to remember a 3-digit code.
He knows the rule that
the first digit is a cube number
the second digit is a factor of 16
the third digit is an odd number.
Liam tries at random a code that matches the rule.
Work out the probability that this is the correct code. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| All three of 1, 8 and 1, 2, 4, 8 and 1, 3, 5, 7, 9 or all three of 2, 4 and 5 | B2 | B1 any two correct do not allow 2, 4 or 5 from an incorrect list of numbers |
| their 2 \(\times\) their 4 \(\times\) their 5 or 40 | M1 | working out the number of possible codes ft their non-zero number of options for each digit implied by \(\dfrac{1}{\text{their } 2} \times \dfrac{1}{\text{their } 4} \times \dfrac{1}{\text{their } 5}\) |
| \(\dfrac{1}{40}\) | A1ft | oe fraction, decimal or percentage ft their non-zero number of options for each digit |
| Alternative method 2 | ||
| All three of \(\dfrac{1}{2}\) and \(\dfrac{1}{4}\) and \(\dfrac{1}{5}\) | B2 | B1 any two correct oe fractions, decimals or percentages do not allow \(\dfrac{1}{2}\), \(\dfrac{1}{4}\) or \(\dfrac{1}{5}\) from an incorrect list of numbers |
| their \(\dfrac{1}{2}\) \(\times\) their \(\dfrac{1}{4}\) \(\times\) their \(\dfrac{1}{5}\) | M1 | oe fractions, decimals or percentages allow their \(\dfrac{1}{2}\) to be 1 their \(\dfrac{1}{4}\) must be \(\lt 1\) their \(\dfrac{1}{5}\) must be \(\lt 1\) |
| \(\dfrac{1}{40}\) | A1ft | oe fraction, decimal or percentage ft their probabilities |
Additional guidance
| If 0 is taken to be a cube number, \(\dfrac{1}{3} \times \dfrac{1}{4} \times \dfrac{1}{5} = \dfrac{1}{60}\) | B1M1A1ft |
| If they only have one cube number, \(1 \times \dfrac{1}{4} \times \dfrac{1}{5} = \dfrac{1}{20}\) | B1M1A1ft |
| 8, 9 and 1, 2, 4, 8 and 1, 3, 5, 7, 9 \(\dfrac{1}{2} \times \dfrac{1}{4} \times \dfrac{1}{5} = \dfrac{1}{40}\) | B1 M1A1ft |
| Ignore conversion attempt after correct answer seen | |
| Allow \(1^3,\ 2^3\) for 1, 8 |