[iron0012]

Here's an explanation that addresses this apparent inconsistency: http://andrewgelman.com/2015/09/30/hot-hand-explanation-agai...

[mcnamaratw]

The linked explanation seems to be that if you do the probability wrong in a certain way, you come up with something below 50%.

Here's one way to reproduce the 40% number they get in the paper. Take a sequence of four flips. Consider five cases:

1. 0 heads. Probability of head following a head=0

2. 1 head. Probability of head following a head=0

3. 2 heads. Probability of head following a head = 1/3

4. 3 heads. Probability of head following a head = 2/3

5. 4 heads. Probability of head following a head = 1

Now if those five cases were equally likely, then what would be the expected number of heads following a head?

Answer: (0 + 0 + 1/3 + 2/3 + 1)/5 = 0.4

Is this what they assume gamblers are using for 'empirical probability'? I can't tell.