On Flipping a Coin and Rosencrantz and Guildenstern

The play “Rosencrantz and Guildenstern are Dead” is a tragicomedy that follows two of the minor characters in “Hamlet” and reveals their perspective of these events. It begins with the two title characters caught in a most unusual coin game. They have been betting on the result of a coin flip and for the last 156 times Rosencrantz has won. Every single time the coin came up as heads. The two characters try as they might to figure out why this is happening.

 

Rosencrantz & Guildenstern Are Dead (film)

Rosencrantz & Guildenstern Are Dead (film) (Photo credit: Wikipedia)

Guildenstern remarks that, “It must be indicative of something besides the redistribution of wealth. A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability. Consider: One, probability is a factor which operates *within* natural forces. Two, probability is *not* operating as a factor. Three, we are now held within un-, sub- or super-natural forces…a spectacular vindication of the principle. That each individual coin spun individually is…as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”

 

As Guildenstern stumbles about for an explanation, he comes across a fundamental part of probability. Each individual coin toss is equally likely to be heads or tails. The first coin toss has no influence on the second one. If the first one is heads then it is still a 50% chance that the next one will be, and a 50% chance that the one after that would be. Even though it would appear that having 156 heads in a row is miraculous, it certainly can happen.

While each of Rosencrantz and Guildenstern’s coin tosses had a 50-50 shot at being heads or tails, the probability of them being all heads is still rare. Consider this model with only three coin tosses. What is the probability that all three are heads? Using a chart all of the possible outcomes can be mapped out.

TOSS 1 TOSS 2 TOSS 3
Heads Heads Heads
Heads Heads Tails
Heads Tails Tails
Heads Tails Heads
Tails Heads Heads
Tails Heads Tails
Tails Tails Heads
Tails Tails Tails

These are all of the possible configurations of a coin toss over three trials. There is only one instance where all three of the outcomes were heads out of the eight different combinations. This means there is a 1/8 chance of three heads happening in three trials, which is equal to 12.5%.

When talking about 156 trials an incredibly large number of rows would be needed to list out all of the possible outcomes. As it turns out, there is a shortcut to help calculate the probability without going through all of that work. The probability of heads for each trial is ½. By multiplying ½ by itself three times (for three trials) the product is 1/8. This is the same as raising ½ to the third power ((½)3 = (1/8)

For 156 heads in 156 trials then the probability would be equivalent to ((½)156. This is ½ multiplied by itself 156 times and the result is an astoundingly small number.

1.09 x 10-47

or

.00000000000000000000000000000000000000000000000109 %

Be careful with how you read this probability. Remember that each individual coin flip has a 50% chance of being heads. The coin does not care what the previous 155 trials were. That being said, it is still 99.99999….% certain that the outcome would be tails, but this is due to how it is being measured. The question is not what is the chance it will be heads, but actually what is the chance there are 156 heads in a row. Even though the percentage is incredibly high that the outcome is tails because having 156 trials of strictly heads is extremely unlikely it is not guaranteed or certain at all. It’s no wonder Rosencrantz and Guildenstern were so baffled.

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