1MA0 Higher November 2014 Paper 1 Q23
23 Paul has 8 cards.
There is a number on each card.
Paul takes at random 3 of the cards.
He adds together the 3 numbers on the cards to get a total \(T\).
Work out the probability that \(T\) is an odd number. (4)
| Answer | Mark | Notes |
|---|---|---|
| \(\dfrac{156}{336}\) | 4 | Method 1 (Combinations for odd T) M1 for one probability for odd T, eg P(2,3,4) = \(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\) or P(2,4,5) = \(\tfrac{1}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\) or P(3,3,5) = \(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\) or P(3,5,5) = \(\tfrac{2}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\) or P(5,5,5) = \(\tfrac{4}{8} \times \tfrac{3}{7} \times \tfrac{2}{6}\) M1 for adding at least two probabilities for odd T, eg \(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6} + \tfrac{1}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\) or \(3\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\right)\) M1 for completely correct method, ie \(6\left(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\right) + 6\left(\tfrac{1}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\right) + 3\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\right) + 3\left(\tfrac{2}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\right) + \left(\tfrac{4}{8} \times \tfrac{3}{7} \times \tfrac{2}{6}\right)\) oe A1 for \(\tfrac{156}{336}\) oe, eg \(\tfrac{13}{28}\) or 0.46(4…) OR Method 2 (Combinations for even T) M1 for one probability for even T, eg P(3,4,5) = \(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\) or P(2,3,3) = \(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\) or P(2,5,5) = \(\tfrac{1}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\) or P(2,3,5) = \(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{4}{6}\) or P(4,5,5) = \(\tfrac{1}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\) or P(3,3,4) = \(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{1}{6}\) M1 for adding at least two probabilities for even T, eg \(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6} + \tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\) or \(3\left(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\right)\) M1 for completely correct method, ie \(1 - \left[6\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{4}{6}\right) + 3\left(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{1}{6}\right) + 3\left(\tfrac{1}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\right) + 6\left(\tfrac{1}{8} \times \tfrac{2}{7} \times \tfrac{4}{6}\right) + 3\left(\tfrac{1}{8} \times \tfrac{4}{7} \times \tfrac{3}{6}\right) + 3\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{1}{6}\right)\right]\) oe A1 for \(\tfrac{156}{336}\) oe, eg \(\tfrac{13}{28}\) or 0.46(4…) OR Method 3 (Combinations of odd and even numbers - odd totals) M1 for one probability for odd T, eg P(E,E,O) = \(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{6}{6}\) or P(O,O,O) = \(\tfrac{6}{8} \times \tfrac{5}{7} \times \tfrac{4}{6}\) M1 for adding at least two probabilities for odd T, eg \(3\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{6}{6}\right)\) or \(\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{6}{6}\right) + \left(\tfrac{6}{8} \times \tfrac{5}{7} \times \tfrac{4}{6}\right)\) M1 for completely correct method, ie \(3\left(\tfrac{2}{8} \times \tfrac{1}{7} \times \tfrac{6}{6}\right) + \left(\tfrac{6}{8} \times \tfrac{5}{7} \times \tfrac{4}{6}\right)\) A1 for \(\tfrac{156}{336}\) oe, eg \(\tfrac{13}{28}\) or 0.46(4…) OR Method 4 (combinations of odd and even numbers - even totals) M1 for probability for even T, ie = \(\tfrac{2}{8} \times \tfrac{6}{7} \times \tfrac{5}{6}\) M1 for adding at least two probabilities for even T, eg \(3\left(\tfrac{2}{8} \times \tfrac{6}{7} \times \tfrac{5}{6}\right)\) M1 for completely correct method, ie \(1 - 3\left(\tfrac{2}{8} \times \tfrac{6}{7} \times \tfrac{5}{6}\right)\) A1 for \(\tfrac{156}{336}\) oe, eg \(\tfrac{13}{28}\) or 0.46(4…) SC (with replacement) For example, M0 M1 for adding at least two probabilities for odd or even T, eg P(E,E,O) = \(\tfrac{2}{8} \times \tfrac{2}{8} \times \tfrac{6}{8}\) or P(O,O,O) = \(\tfrac{6}{8} \times \tfrac{6}{8} \times \tfrac{6}{8}\) M1 for completely correct method, eg \(3\left(\tfrac{2}{8} \times \tfrac{2}{8} \times \tfrac{6}{8}\right) + \left(\tfrac{6}{8} \times \tfrac{6}{8} \times \tfrac{6}{8}\right)\) or \(\tfrac{288}{512}\) oe, eg \(\tfrac{9}{16}\) or 0.56(25) A0 |