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 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.