Higher June 2024 Paper 1 Q16
16 There are 20 sweets in a box.
15 of the sweets are red
5 of the sweets are yellow
Fred takes at random 3 sweets from the box.
Work out the probability that Fred takes at least one sweet of each colour from the box.
(4)
| Scheme | Marks |
|---|---|
\(\dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{5}{18}\left(= \dfrac{35}{228}\right)\) oe or \(\dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{15}{18}\left(= \dfrac{5}{114}\right)\) oe or \(\dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{13}{18}\left(= \dfrac{91}{228}\right)\) oe or \(\dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{3}{18}\left(= \dfrac{1}{114}\right)\) oe | M1 |
\(3 \times \dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{5}{18}\) oe or \(3 \times \dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{15}{18}\) oe or \(\dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{13}{18}\) oe and \(\dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{3}{18}\) oe | M1 |
\(3 \times \dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{5}{18} + 3 \times \dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{15}{18}\) oe or \(\left(\dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{5}{18}\right) + \left(\dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{15}{18}\right) + \left(\dfrac{15}{20} \times \dfrac{5}{19}\right) + \left(\dfrac{5}{20} \times \dfrac{15}{19}\right)\) oe or \(1 - \left(\dfrac{15}{20} \times \dfrac{14}{19} \times \dfrac{13}{18} + \dfrac{5}{20} \times \dfrac{4}{19} \times \dfrac{3}{18}\right)\) oe | M1 |
Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: \(\dfrac{45}{76}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for \(RRY\) or \(YYR\) in any order
or
\(RRR\) or \(YYY\)
Allow equivalent decimals to 2 dp truncated or rounded
Products must be correct (may not be evaluated)
M1: for \((3 \times RRY)\) or \((3 \times YYR)\)
or
\(RRY\) and \(YYR\) (any order)
or
\(RRR\) and \(YYY\)
M1: for a complete method using correct products
A1: oe
0.59(21..) to 2 dp truncated or rounded or 59.(21..)% to 2 sf truncated or rounded
M2 for (first two M1 marks)
\(RY\) and \(YR\)
\(\dfrac{15}{20} \times \dfrac{5}{19}\left(= \dfrac{15}{76}\right)\) oe and
\(\dfrac{5}{20} \times \dfrac{15}{19}\left(= \dfrac{15}{76}\right)\) oe