Radiolab loses at statistics | James Howard Radiolab loses at statistics | James Howard

### James HowardA Mathematician, a Different Kind of Mathematician, and a Statistician

Radiolab has a show on randomness (the embed feature doesn’t work) where they discuss an experiment of flipping a coin 100 times. A string of tails appeared 7 times, which seems to be relatively small. Indeed, given 7 coin tosses, the probability of all 7 coming up tails is

$\frac{1}{2^7} = \frac{1}{128} \approx 0.007812 \text{.}$

As an aside, the probability of all coming up the same side is $\frac{1}{2^6}$ because the initial toss does not matter. All that matters is that all subsequent tosses match.

The discussion sets up an experiment where one group flips a coin 100 times and a second group says they flip a coin 100 times and each reports the results. A statistics professor can then tell the difference because the group that actually flipped the coin got a string of 7 tails which seems unlikely, but is more likely than you think. The show notes that the probability of getting 7 tails is 1 in 6 because there are 14 groups of 7 in 100. Which in one sense is true. But with overlap, there are 94 groups of 7 any one of which can be all tails. So the probability of getting a string of exactly 7 tails is really

$1 - \big(\frac{127}{128}\big)^{94} \approx 0.521576 \text{.}$

Or just more than half of all sets of 100 flips will contain a string of 7 tails. And if we can allow that we don’t care about whether the string of 7 is either heads or tails, the probability jumps to more than three-quarters:

$1 - \big(\frac{63}{64}\big)^{94} \approx 0.772440 \text{.}$

These are all a lot more than the 1 in 6 Radiolab claims.