Higher November 2020 Paper 2 Q27
27 These 20 discs are in a bag.
Two of the discs are taken at random from the bag.
Work out the probability that the first disc has a smaller number than the second disc. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{4}{20} \times \dfrac{16}{19}\) or \(\dfrac{64}{380}\) or \(\dfrac{16}{95}\) or \(\dfrac{6}{20} \times \dfrac{10}{19}\) or \(\dfrac{60}{380}\) or \(\dfrac{3}{19}\) or \(\dfrac{7}{20} \times \dfrac{3}{19}\) or \(\dfrac{21}{380}\) | M1 | oe fractions or decimals condone \(\dfrac{4}{20} \times \dfrac{16}{20}\) etc |
| Any 2 of \(\dfrac{4}{20} \times \dfrac{16}{19}\) or \(\dfrac{64}{380}\) or \(\dfrac{16}{95}\) and \(\dfrac{6}{20} \times \dfrac{10}{19}\) or \(\dfrac{60}{380}\) or \(\dfrac{3}{19}\) and \(\dfrac{7}{20} \times \dfrac{3}{19}\) or \(\dfrac{21}{380}\) | M1dep | oe fractions or decimals |
| \(\dfrac{4}{20} \times \dfrac{16}{19} + \dfrac{6}{20} \times \dfrac{10}{19} + \dfrac{7}{20} \times \dfrac{3}{19}\) or \(\dfrac{64}{380} + \dfrac{60}{380} + \dfrac{21}{380}\) | M1dep | oe fractions or decimals eg \(\dfrac{16}{95} + \dfrac{3}{19} + \dfrac{21}{380}\) |
| \(\dfrac{145}{380}\) or \(\dfrac{29}{76}\) or [0.381, 0.382] or [38.1%, 38.2%] | A1 | accept 0.38 or 38% with full working SC2 \(\dfrac{145}{400}\) or \(\dfrac{29}{80}\) or 0.3625 or 36.25% |
| Alternative method 2 | ||
| \(\dfrac{6}{20} \times \dfrac{4}{19}\) or \(\dfrac{24}{380}\) or \(\dfrac{6}{95}\) or \(\dfrac{7}{20} \times \dfrac{10}{19}\) or \(\dfrac{70}{380}\) or \(\dfrac{7}{38}\) or \(\dfrac{3}{20} \times \dfrac{17}{19}\) or \(\dfrac{51}{380}\) | M1 | oe fractions or decimals condone \(\dfrac{6}{20} \times \dfrac{4}{20}\) etc |
| Any 2 of \(\dfrac{6}{20} \times \dfrac{4}{19}\) or \(\dfrac{24}{380}\) or \(\dfrac{6}{95}\) and \(\dfrac{7}{20} \times \dfrac{10}{19}\) or \(\dfrac{70}{380}\) or \(\dfrac{7}{38}\) and \(\dfrac{3}{20} \times \dfrac{17}{19}\) or \(\dfrac{51}{380}\) | M1dep | oe fractions or decimals |
| \(\dfrac{6}{20} \times \dfrac{4}{19} + \dfrac{7}{20} \times \dfrac{10}{19} + \dfrac{3}{20} \times \dfrac{17}{19}\) or \(\dfrac{24}{380} + \dfrac{70}{380} + \dfrac{51}{380}\) | M1dep | oe fractions or decimals eg \(\dfrac{6}{95} + \dfrac{7}{38} + \dfrac{51}{380}\) |
| \(\dfrac{145}{380}\) or \(\dfrac{29}{76}\) or [0.381, 0.382] or [38.1%, 38.2%] | A1 | accept 0.38 or 38% with full working SC2 \(\dfrac{145}{400}\) or \(\dfrac{29}{80}\) or 0.3625 or 36.25% |
| Alternative method 3 | ||
| \(\dfrac{6}{20} \times \dfrac{15}{19}\) or \(\dfrac{90}{380}\) or \(\dfrac{9}{38}\) or \(\dfrac{7}{20} \times \dfrac{9}{19}\) or \(\dfrac{63}{380}\) or \(\dfrac{3}{20} \times \dfrac{2}{19}\) or \(\dfrac{6}{380}\) or \(\dfrac{3}{190}\) | M1 | oe fractions or decimals condone \(\dfrac{6}{20} \times \dfrac{15}{20}\) etc |
| Any 2 of \(\dfrac{6}{20} \times \dfrac{15}{19}\) or \(\dfrac{90}{380}\) or \(\dfrac{9}{38}\) and \(\dfrac{7}{20} \times \dfrac{9}{19}\) or \(\dfrac{63}{380}\) and \(\dfrac{3}{20} \times \dfrac{2}{19}\) or \(\dfrac{6}{380}\) or \(\dfrac{3}{190}\) | M1dep | oe fractions or decimals |
| \(1 - \dfrac{4}{20} - \dfrac{6}{20} \times \dfrac{15}{19} - \dfrac{7}{20} \times \dfrac{9}{19} - \dfrac{3}{20} \times \dfrac{2}{19}\) or \(1 - \dfrac{4}{20} - \dfrac{90}{380} - \dfrac{63}{380} - \dfrac{6}{380}\) | M1dep | oe fractions or decimals eg \(1 - \dfrac{1}{5} - \dfrac{9}{38} - \dfrac{63}{380} - \dfrac{3}{190}\) |
| \(\dfrac{145}{380}\) or \(\dfrac{29}{76}\) or [0.381, 0.382] or [38.1%, 38.2%] | A1 | accept 0.38 or 38% with full working SC2 \(\dfrac{145}{400}\) or \(\dfrac{29}{80}\) or 0.3625 or 36.25% |
| Alternative method 4 | ||
| \(\dfrac{7}{20} \times \dfrac{16}{19}\) or \(\dfrac{112}{380}\) or \(\dfrac{28}{95}\) or \(\dfrac{6}{20} \times \dfrac{9}{19}\) or \(\dfrac{54}{380}\) or \(\dfrac{27}{190}\) or \(\dfrac{4}{20} \times \dfrac{3}{19}\) or \(\dfrac{12}{380}\) or \(\dfrac{3}{95}\) | M1 | oe fractions or decimals condone \(\dfrac{7}{20} \times \dfrac{16}{20}\) etc |
| Any 2 of \(\dfrac{7}{20} \times \dfrac{16}{19}\) or \(\dfrac{112}{380}\) or \(\dfrac{28}{95}\) and \(\dfrac{6}{20} \times \dfrac{9}{19}\) or \(\dfrac{54}{380}\) or \(\dfrac{27}{190}\) and \(\dfrac{4}{20} \times \dfrac{3}{19}\) or \(\dfrac{12}{380}\) or \(\dfrac{3}{95}\) | M1dep | oe fractions or decimals |
| \(1 - \dfrac{3}{20} - \dfrac{7}{20} \times \dfrac{16}{19} - \dfrac{6}{20} \times \dfrac{9}{19} - \dfrac{4}{20} \times \dfrac{3}{19}\) or \(1 - \dfrac{3}{20} - \dfrac{112}{380} - \dfrac{54}{380} - \dfrac{12}{380}\) | M1dep | oe fractions or decimals eg \(1 - \dfrac{3}{20} - \dfrac{28}{95} - \dfrac{27}{190} - \dfrac{3}{95}\) |
| \(\dfrac{145}{380}\) or \(\dfrac{29}{76}\) or [0.381, 0.382] or [38.1%, 38.2%] | A1 | accept 0.38 or 38% with full working SC2 \(\dfrac{145}{400}\) or \(\dfrac{29}{80}\) or 0.3625 or 36.25% |
| Alternative method 5 | ||
| \(4 \times 16\) or \(6 \times 10\) or \(7 \times 3\) or \(3 \times 17\) or \(7 \times 10\) or \(6 \times 4\) | M1 | oe eg 64 or 60 or 21 or 51 or 70 or 24 |
| Any 2 of \(4 \times 16\) and \(6 \times 10\) and \(7 \times 3\) or any 2 of \(3 \times 17\) and \(7 \times 10\) and \(6 \times 4\) | M1dep | oe implied by 145 |
| \(\dfrac{4 \times 16 + 6 \times 10 + 7 \times 3}{20 \times 19}\) or \(\dfrac{3 \times 17 + 7 \times 10 + 6 \times 4}{20 \times 19}\) | M1dep | oe |
| \(\dfrac{145}{380}\) or \(\dfrac{29}{76}\) or [0.381, 0.382] or [38.1%, 38.2%] | A1 | accept 0.38 or 38% with full working SC2 \(\dfrac{145}{400}\) or \(\dfrac{29}{80}\) or 0.3625 or 36.25% |
Additional guidance
| Ignore simplification or conversion attempt after correct answer seen | |
| For M marks accept oe decimals rounded to 2 dp or better | |
| Select the scheme that favours the student for the first 2 M marks even if not subsequently used | |
| Using \(\dfrac{4}{20} \times \dfrac{16}{20}\) etc can score M1M0M0A0 or SC2 | |
| Do not award marks if a fraction comes from an incorrect method eg Alt 1 \(\dfrac{4}{20} \times \dfrac{15}{19} = \dfrac{3}{19}\) | M0 |