Higher November 2024 Unit 2 Q18
18. A bag contains 3 red counters, 5 blue counters and 7 yellow counters.
Sali takes three counters at random from the bag. These counters are not replaced.
Calculate the probability that at least two of the counters are the same colour. [4]
| Scheme | Marks | Comments |
|---|---|---|
| ( 1− P(all 3 colours are chosen) = ) \(1 - 6 \times \dfrac{3}{15} \times \dfrac{5}{14} \times \dfrac{7}{13}\) | M3 | M2 for \(6 \times \dfrac{3}{15} \times \dfrac{5}{14} \times \dfrac{7}{13}\) \(\left(= \dfrac{3}{13} = \dfrac{630}{2730}\right)\) M1 for \(1 - \dfrac{3}{15} \times \dfrac{5}{14} \times \dfrac{7}{13}\) \(\left(= \dfrac{25}{26} = \dfrac{2625}{2730}\right)\) |
| \(= \dfrac{10}{13}\left(= \dfrac{2100}{2730}\right)\) | A1 | CAO ISW Accept a decimal answer of 0·76(9…) or 0·77. If no marks award SC1 for one of the following:
|
Alternative method 1
| Scheme | Marks | Comments |
|---|---|---|
| May be seen in stages. | ||
| ( P(2 Red & 1 Not Red) OR P(2 Blue & 1 Not Blue) OR P(2 Yellow & 1 Not Yellow) =) \(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{12}{13} +\) \(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{10}{13} +\) \(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{8}{13}\) \(\left(= \dfrac{304}{455} = \dfrac{1824}{2730}\right)\) | M2 | M1 for: \(\left(\dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{12}{13}\right) + \left(\dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{10}{13}\right) + \left(\dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{8}{13}\right)\) \(\left(= \dfrac{12}{455} + \dfrac{20}{273} + \dfrac{8}{65} = \dfrac{304}{1365}\right)\) OR \(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{12}{13}\) \(\left(= \dfrac{36}{455} = \dfrac{216}{2730}\right)\) OR \(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{10}{13}\) \(\left(= \dfrac{20}{91} = \dfrac{600}{2730}\right)\) OR \(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{8}{13}\) \(\left(= \dfrac{24}{65} = \dfrac{1008}{2730}\right)\) |
| (P(3 Red) OR P(3 Blue) OR P(3 Yellow) =) \(\left(\dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{1}{13}\right) + \left(\dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\right) + \left(\dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\right)\) \(\left(= \dfrac{46}{455} = \dfrac{276}{2730}\right)\) | M1 | Award M1M1 for an answer of \(\dfrac{34}{105}\) from sight of: P(R,R,other)+ P(B,B,other)+ P(Y,Y,other) OR P(R,R)+ P(B,B)+ P(Y,Y) |
| (P(at least 2 counters the same colour) =) \(\dfrac{10}{13}\left(= \dfrac{2100}{2730}\right)\) | A1 | CAO ISW Accept a decimal answer of 0·76(9…) or 0·77. If no marks award SC1 for one of the following:
|
Alternative method 2
| Scheme | Marks | Comments |
|---|---|---|
| May be seen in stages. | ||
| ( P(2 Red & 1 Blue) OR ( P(2 Red & 1 Yellow) OR P(2 Blue & 1 Red) OR P(2 Blue & 1 Yellow) OR P(2 Yellow & 1 Red) OR P(2 Yellow & 1 Blue) =) \(\left(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{5}{13}\right) + \left(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{7}{13}\right) +\) \(\left(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\right) + \left(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{7}{13}\right) +\) \(\left(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{3}{13}\right) + \left(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\right)\) \(\left(= \dfrac{304}{455} = \dfrac{1824}{2730}\right)\) | M2 | M1 for: \(\left(\dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{5}{13}\right) + \left(\dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{7}{13}\right) +\) \(\left(\dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\right) + \left(\dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{7}{13}\right) +\) \(\left(\dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{3}{13}\right) + \left(\dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\right)\) \(\left(= \dfrac{304}{1365} = \dfrac{608}{2730}\right)\) OR \(\left(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{5}{13}\right) + \left(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{7}{13}\right)\left(= \dfrac{36}{455} = \dfrac{216}{2730}\right)\) OR \(\left(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\right) + \left(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{7}{13}\right)\left(= \dfrac{20}{91} = \dfrac{600}{2730}\right)\) OR \(\left(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{3}{13}\right) + \left(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\right)\left(= \dfrac{24}{65} = \dfrac{1008}{2730}\right)\) |
| (P(3 Red) OR P(3 Blue) OR P(3 Yellow) =) \(\left(\dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{1}{13}\right) + \left(\dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\right) + \left(\dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\right)\) \(\left(= \dfrac{46}{455} = \dfrac{276}{2730}\right)\) | M1 | |
| (P(at least 2 counters the same colour) =) \(\dfrac{10}{13}\left(= \dfrac{2100}{2730}\right)\) | A1 | CAO ISW Accept a decimal answer of 0·76(9…) or 0·77. If no marks award SC1 for one of the following:
|
Alternative method 3
| Scheme | Marks | Comments |
|---|---|---|
| ( P(at least 2 Red) = P(2 Red & 1 Not Red) + P(3 Red)=) \(3 \times \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{12}{13} + \dfrac{3}{15} \times \dfrac{2}{14} \times \dfrac{1}{13}\) \(\left(\dfrac{37}{455} = \dfrac{222}{2730}\right)\) | M1 | Calculations may be seen in stages e.g. P(2 Red & 1 Not Red) = P(2 Red & 1 Blue) + P(2 Red & 1 Yellow) |
| ( P(at least 2 Blue) = P(2 Blue & 1 Not Blue) + P(3 Blue)=) \(3 \times \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{10}{13} + \dfrac{5}{15} \times \dfrac{4}{14} \times \dfrac{3}{13}\) \(\left(\dfrac{22}{91} = \dfrac{660}{2730}\right)\) | M1 | |
| ( P(at least 2 Yellow) = P(2 Yellow & 1 Not Yellow) + P(3 Yellow)=) \(3 \times \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{8}{13} + \dfrac{7}{15} \times \dfrac{6}{14} \times \dfrac{5}{13}\) \(\left(\dfrac{203}{455} = \dfrac{1218}{2730}\right)\) | M1 | |
| (P(at least 2 counters the same colour) =) \(\dfrac{10}{13}\left(\dfrac{2100}{2730}\right)\) | A1 | CAO ISW Accept a decimal answer of 0·76(9…) or 0·77. If no marks award SC1 for one of the following:
|