2002 AMC 8 Problem 21

Attempt Problem 21 of the 2002 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

21.

Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is
516\dfrac{5}{16}
38\dfrac{3}{8}
12\dfrac{1}{2}
58\dfrac{5}{8}
1116\dfrac{11}{16}
Answer: E
Concepts:binomial probabilitycombinations
Small hint:
At least as many heads as tails means 2,2, 3,3, or 44 heads.
Big hint:
Count outcomes by choosing which tosses are heads.
Solution:
There are 24=162^4=16 equally likely outcomes for four coin tosses.
At least as many heads as tails means getting 2,2, 3,3, or 44 heads. The number of such outcomes is (42)+(43)+(44)=6+4+1=11. \begin{aligned} &\binom{4}{2}+\binom{4}{3}+\binom{4}{4} \\ &\quad {}=6+4+1=11. \end{aligned}
Thus the probability is 1116.\dfrac{11}{16}.
Thus, E is the correct answer.
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