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Reuben is playing a matching game with these cards - OCR - GCSE Maths - Question 11 - 2017 - Paper 1
Question 11
Reuben is playing a matching game with these cards. He turns the cards over and shuffles them. Reuben takes a card and keeps it. He then takes a second card. If the... show full transcript
Worked Solution & Example Answer:Reuben is playing a matching game with these cards - OCR - GCSE Maths - Question 11 - 2017 - Paper 1
Step 1
Complete this tree diagram to show the probabilities for each card picked in the game.
Answer
To complete the tree diagram, we need to calculate the probabilities for each card selection.
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First Card: Reuben has 4 circle cards and 5 square cards out of a total of 9 cards. Therefore:
- Probability of picking a Circle first: ( P(Circle) = \frac{4}{9} )
- Probability of picking a Square first: ( P(Square) = \frac{5}{9} )
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Second Card: Based on what he picks first:
- If the first card is a Circle (4 cards remain), there will be 3 circles and 5 squares left:
- Probability of picking Circle second: ( P(Circle ; | ; Circle) = \frac{3}{8} )
- Probability of picking Square second: ( P(Square ; | ; Circle) = \frac{5}{8} )
- If the first card is a Square (5 cards remain), there will be 4 circles and 4 squares left:
- Probability of picking Circle second: ( P(Circle ; | ; Square) = \frac{4}{8} = \frac{1}{2} )
- Probability of picking Square second: ( P(Square ; | ; Square) = \frac{4}{8} = \frac{1}{2} )
- If the first card is a Circle (4 cards remain), there will be 3 circles and 5 squares left:
The completed tree diagram probabilities are:
- First Card: Circle - Probability ( \frac{4}{9} )
- Second Card: Circle - Probability ( \frac{3}{8} )
- Second Card: Square - Probability ( \frac{5}{8} )
- First Card: Square - Probability ( \frac{5}{9} )
- Second Card: Circle - Probability ( \frac{4}{8} )
- Second Card: Square - Probability ( \frac{4}{8} )
Step 2
What is the probability that Reuben wins the game?
Answer
Reuben wins if he picks different cards in his two selections.
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If the first card is Circle and the second card is Square:
- Probability: ( P(Circle) \times P(Square ; | ; Circle) = \frac{4}{9} \times \frac{5}{8} = \frac{20}{72} = \frac{5}{18} )
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If the first card is Square and the second card is Circle:
- Probability: ( P(Square) \times P(Circle ; | ; Square) = \frac{5}{9} \times \frac{4}{8} = \frac{20}{72} = \frac{5}{18} )
To find the total probability that Reuben wins, we sum the probabilities of the two winning scenarios:
[ P(Win) = P(Circle ; first, ; Square ; second) + P(Square ; first, ; Circle ; second) ] [ P(Win) = \frac{5}{18} + \frac{5}{18} = \frac{10}{18} = \frac{5}{9} ]
Thus, the probability that Reuben wins the game is ( \frac{5}{9} ).