Problem 30 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyor-process-add
A nickel and a dime are tossed at the same time. Find the probability that (a) both show heads; (b) both show tails; (c) one shows heads and the other shows tails.
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Answer: (a) 1/4; (b) 1/4; (c) 1/2
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Hint 1 of 3
The two coins are independent, so 'and' results multiply (each coin is heads with probability \(1/2\)).
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Hint 2 of 3
(a) and (b) are single 'and' outcomes, each \((1/2)(1/2)\). (c) can happen two ways: nickel H and dime T, OR nickel T and dime H.
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Hint 3 of 3
(c): add the two ways, each \((1/2)(1/2)\).
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Approach: AND for each coin, OR across the two ways for part (c)
- Each coin is heads or tails with probability \(\frac{1}{2}\), and the coins are independent, so multiply for 'and'.
- (a) Both heads: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\). (b) Both tails: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).
- (c) One head, one tail — two separate ways (OR, add): (nickel H, dime T) or (nickel T, dime H), so \(\frac{1}{4} + \frac{1}{4} = \frac{1}{2}\).
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