Chance News 106
Quotations
Forsooth
You really are old as you feel
"Some of the subjects 'aged physiologically not at all [while] at the other extreme there were folks aging two to three times as much.'"
Submitted by Chris Andrews
More on the hot hand
In Chance News 105, the last item was titled Does Selection bias explain the hot hand?. It described how in their July 6 article, Miller and Sanjurjo assert that a way to determine the probability of a heads following a heads in a fixed sequence, you may calculate the proportion of times a head is followed by a head for each possible sequence and then compute the average proportion, giving each sequence an equal weighting on the grounds that each possible sequence is equally likely to occur. I agree that each possible sequence is equally likely to occur. But I assert that it is illegitimate to weight each sequence equally because some sequences have more chances for a head to follow a second head than others.
Let us assume, as Miller and Sanjurjo do, that we are considering the 14 possible sequences of four flips containing at least one head in the first three flips. A head is followed by another head in only one of the six sequences (see below) that contain only one head that could be followed by another, making the probability of a head being followed by another 1/6 for this set of six sequences.
TTHT Heads follows heads 0 time. THTT Heads follows heads 0 times HTTT Heads follows heads 0 times TTHH Heads follows heads 1 time THTH Heads follows heads 0 times HTTH Heads follows heads 0 times
A head is followed by another head six times in the six sequences (see below) that contain two heads that could be followed by another head, making the probability of a head being followed by another 6/12 = 1/2 for this set of six sequences.
THHT Heads follows heads 1 time HTHT Heads follows heads 0 times HHTT Heads follows heads 1 time THHH Heads follows heads 2 times HTHH Heads follows heads 1 time HHTH Heads follows heads 1 time
A head is followed by another head five times in the six sequences (see below) that contain three heads that could be followed by another head, making the probability of a head being followed by another 5/6 this set of two sequences.
HHHT Heads follows heads 2 times HHHH Heads follows heads 3 times
An unweighted average of the 14 sequences gives
<math>[(6 \times 1/6) + (6 \times 1/2) + (2 \times 5/6)]/14 = [17/3]/14 = 0.405</math>
which is what Miller and Sanjurjo report. A weighted average of the 14 sequences gives
<math> [(1)(6 \times 1/6) + (2)(6 \times 1/2) + (3)(2 \times 5/6)]/[(1 \times 6) + (2 \times 6) + (3 \times 2)] = [1 + 6 +5]/[6 + 12 + 6] = 12/24 = 0.50.</math>
Using an unweighted average instead of a weighted average is the pattern of reasoning underlying the statistical artifact known as Simpson’s paradox. And as is the case with Simpson’s paradox, it leads to faulty conclusions about how the world works.
Submitted by Jeff Eiseman, University of Massachusetts