Problem 4 · AMC 8 Stretch
Core
Counting & Probability
account-for-all-possibilitieslogical-reasoning
A flower has 6 petals. Each petal opens with probability \(\tfrac12\) (like a fair coin: heads = open, tails = closed), and the petals are independent. What is the probability that EXACTLY 2 of the 6 petals open?
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Answer: 15/64 (about 0.23)
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Hint 1 of 4
There are \(2^6 = 64\) equally likely open/closed patterns (each petal a fair coin). So you can find the probability by counting.
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Hint 2 of 4
You need patterns with exactly 2 petals open out of 6. That's the same as asking: in how many ways can you CHOOSE which 2 of the 6 petals are the open ones?
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Hint 3 of 4
Count the choices of 2 petals from 6: \(\tfrac{6 \times 5}{2} = 15\) ways.
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Approach: Count favorable patterns over all equally likely patterns
- Because each petal is like a fair coin, all \(2^6 = 64\) open/closed patterns are equally likely, so \(P(\text{exactly 2 open}) = \dfrac{\text{patterns with exactly 2 open}}{64}\).
- Patterns with exactly 2 open means choosing which 2 of the 6 petals are open. The number of ways to pick 2 of 6 is \(\tfrac{6 \times 5}{2} = 15\) (6 choices for the first, 5 for the second, divide by 2 since order doesn't matter).
- So there are 15 good patterns out of 64: \(P(\text{exactly 2 open}) = \dfrac{15}{64} \approx 0.23\).
- The probability is \(\dfrac{15}{64}\).
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