Expected Value: Coins

Calculate expected value for coin scenarios

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Expected Number of Heads in n Flips

What does "expected number of heads" mean?

If you flip a coin n times many times over, and count the heads each time, the average of all those head counts is the expected value. You won't necessarily get exactly E(X) heads on any one trial!

The clever trick: Indicator Variables

Instead of listing all possible head counts (0, 1, 2, ..., n), we think of each flip as a separate "indicator" that's 1 for heads and 0 for tails. Total heads = sum of these indicators.

Question

If you flip a fair coin n times, how many heads do you expect to get?

Solution: Indicator Variable Method

Define Indicator Variables

Let = 1 if flip i is heads, 0 otherwise.

Total heads:

Find E(X_i) for Each Flip

Apply Linearity of Expectation

General Formula

Examples

10 flips

5 heads

100 flips

50 heads

1000 flips

500 heads

Extension: Biased Coin

If the coin has P(H) = p (not necessarily 1/2):

Example: A biased coin with P(H) = 0.7, flipped 20 times:
E(heads) = 20 × 0.7 = 14 heads expected.

Key Insight: Linearity Power

The indicator variable method with linearity of expectation is extremely powerful. It works even when the flips are dependent (though for a fair coin they are independent). This technique simplifies many counting problems.