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Jack picks a counter at random and then replaces it - OCR - GCSE Maths - Question 7 - 2019 - Paper 5
Question 7
Jack picks a counter at random and then replaces it. Jack then picks a second counter at random. (a) Complete the tree diagram. First pick Second pick 4 7 Red 4... show full transcript
Worked Solution & Example Answer:Jack picks a counter at random and then replaces it - OCR - GCSE Maths - Question 7 - 2019 - Paper 5
Step 1
Complete the tree diagram.
Answer
To complete the tree diagram, we need to calculate the probabilities of each outcome based on Jack's first pick and the total counters.
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First pick: The bag has a total of 7 counters (4 red and 3 blue).
- Probability of picking a red counter: ( \frac{4}{7} )
- Probability of picking a blue counter: ( \frac{3}{7} )
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Second pick: Since the counter is replaced, the probabilities remain the same for the second pick.
- If the first pick was red (4/7):
- Probability of red again: ( \frac{4}{7} )
- Probability of blue: ( \frac{3}{7} )
- If the first pick was blue (3/7):
- Probability of red: ( \frac{4}{7} )
- Probability of blue: ( \frac{3}{7} )
- If the first pick was red (4/7):
The completed tree diagram will contain:
- First pick Red: 4/7, Second pick Red: 4/7
- First pick Red: 4/7, Second pick Blue: 3/7
- First pick Blue: 3/7, Second pick Red: 4/7
- First pick Blue: 3/7, Second pick Blue: 3/7
Step 2
Work out the probability that Jack picks two red counters.
Answer
To find the probability that Jack picks two red counters:
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First, calculate the probability of picking a red counter on the first pick: ( P(R) = \frac{4}{7} ).
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Next, since the counter is replaced, the probability of picking a red counter again on the second pick is the same: ( P(R) = \frac{4}{7} ).
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The combined probability of both events (picking red first and picking red second) can be calculated using the formula for independent events:
[ P(RR) = P(R) \times P(R) = \frac{4}{7} \times \frac{4}{7} = \frac{16}{49} ]
Thus, the probability that Jack picks two red counters is ( \frac{16}{49} ).