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counting and probability intermediate
Problem
A fair coin is flipped 7 times. What is the probability that at least 5 of the flips come up heads?
Solution
First, we count the number of total outcomes. Each toss has 2 possibilities - heads or tails - so the 7 tosses have possible outcomes.
To count the number of outcomes with at least 5 heads, we need to use casework.
Case 1: 5 heads. To count the number of ways that 5 heads can come up, we simply need to choose 5 of the 7 tosses to be heads (the other 2 tosses will then automatically be tails). So this can be done in ways.
Case 2: 6 heads. Here we have to choose 6 of the tosses to be heads; this can be done in ways.
Case 3: 7 heads. There's only 1 way to do this -- all 7 tosses must be heads.
So there are successful outcomes, hence the probability is .
To count the number of outcomes with at least 5 heads, we need to use casework.
Case 1: 5 heads. To count the number of ways that 5 heads can come up, we simply need to choose 5 of the 7 tosses to be heads (the other 2 tosses will then automatically be tails). So this can be done in ways.
Case 2: 6 heads. Here we have to choose 6 of the tosses to be heads; this can be done in ways.
Case 3: 7 heads. There's only 1 way to do this -- all 7 tosses must be heads.
So there are successful outcomes, hence the probability is .
Final answer
\frac{29}{128}