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4MA1/1H Nov 2021, Question 15: Tree Diagrams for Independent Events

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 8 marks

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The question, in short

Two chess games with win probability 2/9 and draw 4/9, independent. Complete a tree diagram, then find probabilities that the players' points are equal after 2 games and after a third game.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Fill in the missing branch probabilities on the two-stage tree diagram for the first and second games.

  1. 1The three outcomes add to 1: P(Magnus loses) = 1 − 2/9 − 4/9 = 3/9
  2. 2First game: Magnus loses = 3/9
  3. 3Games are independent, so each second-game set is 2/9, 4/9, 3/9

AnswerFirst game: Magnus loses 3/9. Second game, on all three pairs of branches: Magnus wins 2/9, Game drawn 4/9, Magnus loses 3/9

Part (b) · 3 marks

Using 2 points for a win and 1 each for a draw, find the probability Magnus and Garry have equal points after 2 games.

  1. 1Equal points after 2 games: both drawn, or one win and one loss each way
  2. 2P(D, D) = 4/9 × 4/9 = 16/81
  3. 3P(W, L) + P(L, W) = 2/9 × 3/9 + 3/9 × 2/9 = 12/81
  4. 416/81 + 12/81 = 28/81

Answer28/81

Part (c) · 3 marks

Find the probability that the two players have equal points after a third game is added.

  1. 1Equal points after 3 games: three draws, or one win, one loss and one draw
  2. 2P(D, D, D) = (4/9)³ = 64/729
  3. 3Win, loss, draw in any of 6 orders: 6 × 2/9 × 3/9 × 4/9 = 144/729
  4. 464/729 + 144/729 = 208/729

Answer208/729

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