Flipping coins

Determine how many times you must flip a fair coin so that:

Variant
The expected number of heads is 5.
Solution
One flip gives us \(\frac12\) head on average, so we must flip the coin ten times. (This follows from the linearity of expectation.)
Variant
The probability of at least 5 heads is \(\frac12\).
Solution
If we flip the coin nine times, then we can form pairs from the possible results by overturning all coins in a result set. In each pair one event will have at least 5 heads and the second will have fewer than five heads. So for this number of coins the probability of the required event and its complement are the same, so the probability is \(\frac12\).