2002 AMC 8 Problems/Problem 21
Problem
Harold tosses a coin four times. The probability that he gets at least as many heads as tails is
Solution
Case 1: There are two heads and two tails. There are ways to choose which two tosses are heads, and the other two must be tails.
Case 2: There are three heads, one tail. There are ways to choose which of the four tosses is a tail.
Case 3: There are four heads and no tails. This can only happen way.
There are a total of possible configurations, giving a probability of .
Solution 2 (fastest)
We want the probability of at least two heads out of . We can do this a faster way by noticing that the probabilities are symmetric around two heads. Define as the probability of getting heads on rolls. Now our desired probability is . We can easily calculate because there are ways to get heads and tails, and there are total ways to flip these coins, giving , and plugging this in gives us . ~chrisdiamond10
Solution 3
You can use complimentary counting to solve this problem. Because we are trying to figure out the probability that Harold gets at least as many heads as tails, when using complimentary counting, we want to find the probability that there are MORE tails than heads. That are the cases when there are tails and head, and all tails. For the first case ( tails and head), there are cases that it works.




For the second case (all tails), there is case that works. Which is, of course .
So in total, there is cases that there are more tails then heads out of possibilities. Since we are doing complimentary counting, we need to subtract from one. Hence the answer is .
Video Solution
https://www.youtube.com/watch?v=4vLTPszBLeg ~David
See Also
2002 AMC 8 (Problems • Answer Key • Resources)  
Preceded by Problem 20 
Followed by Problem 22  
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25  
All AJHSME/AMC 8 Problems and Solutions 
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