This article makes for a good companion piece to my post on dice probabilities.
Soon we find out that Guildenstern [in the play Rosencrantz and Guildenstern are dead] has flipped 76 coins, and all of them have come up heads. “A weaker man,” he remarks, “might be moved to re-examine his faith, if in nothing else at least in the law of probability.”
The article goes on to conclude that it’s unlikely that anyone has actually flipped 76 heads in a row. The logic is as follows:
If a coin is flipped 76 times there are 2 to the power of 76 different possible outcomes. That’s over 75 sextillion possibilities. (If it helps, you can think of these flips as a binary number composed of 76 bits.) So the chance of flipping 76 heads is incredibly rare. 1 in 75 sextillion rare.
However, a fun paradox arises: Can I not flip a coin 76 times and then state that based its rarity (1 in 75 sextillion) that I doubt this particular sequence has ever been flipped, even though it just has?!
Perhaps, but it’s important to note that this only works for ordered sequences of coin flips, not the ratio of heads to tails within. This distinction is important because many different sequences can lead to the same heads to tails ratio.
For example, if on my first flip I get tails followed by 75 heads, this is a 1 in 75 sextillion sequence. Compare this with the probability of flipping one tails anywhere amongst 75 heads. Since the lone tails could appear on any of the 76 flips, the probability is 76 in 75 sextillion.
As the heads to tails ratio approaches 50/50, the odds get much better. A sequence containing 38 heads and 38 tails should be tossed once every 11 attempts (approximately). This is due to the large number of sequences within the 75 sextillion that contain an equal number of heads and tails.