Expected Value: Coins

Calculate expected value for coin scenarios

Expected Flips Until First Heads

STOP

Flip a fair coin until you get heads. On average, how many flips will this take?

The number of trials until first success follows a geometric distribution.

If probability of success (heads) on each trial is p:

To get first heads on flip k: need k-1 tails, then heads.

For , we have

With :

Answer

E(X) = 2 flips

On average, you need 2 flips to get your first heads.

Flips (k)	Sequence	P(X = k)	Cumulative
1	H		50.00%
2	TH		75.00%
3	TTH		87.50%
4	TTTH		93.75%
5	TTTTH		96.88%
6	TTTTTH		98.44%
7+	TTTTTT...H	1.56%	100%

For a coin with P(heads) = p:

p = 0.5 (fair)

2 flips

p = 0.25

4 flips

p = 0.1

10 flips

The geometric distribution is memoryless:

If you have already flipped 10 tails in a row, the expected number of additional flips until heads is still 2, not less! Each flip is independent.