A Mathematical Paradox About Probabilities

Something to think about:

Since the coin flips are independent, and assuming the coin is fair, the probability that ten coin flips land heads is:

$$P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H) \cdot P(H)\cdot P(H)=(0.5)^{10}$$

The probability that nine coin flips land heads and the tenth lands tails is:

$$P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H)\cdot P(H) \cdot P(H)\cdot P(T)=(0.5)^{10}$$

The probabilities are the same! So he had equal chances of dying or not dying.


All the answers provided explain why there is no "paradox." I would like to provide you an answer that is rather intuitive (hopefully) than formal.

Your question is a good example of what is known as "gamblers' fallacy" (see, Croson and Sundali (2005)). The fallacy occurs when one wrongly assumes that a bin from which a draw is made is finite. In your example, think of having a head as drawing a blue ball from a bin and having a tail as drawing a red ball from the same bin, and the bin contains countably infinite balls half of which are blue and the other half red. Notice that if you draw $9$ blue balls from the bin in a row, the probability of drawing another ball is still $\frac{1}{2}$ since there are still infinitely many blue balls left. this is true even if you draw $1000$ blue balls (in fact, any finite number of balls) from the bin in a row -- meaning that the probability that $1001$st ball is blue is still $\frac{1}{2}$. This is the case with the coin toss: even if heads come up $1000$ times in a row, the probability that a head comes up in $1001$st flip is $\frac{1}{2}$. The reason why you think you have a paradox is that you are mistakenly assuming that you are drawing balls from some bin that contains finite number of balls.


The answers given so far are all theoretical. Let's be practical.

Exercise #1:

Go get a coin right now, and a piece of paper.

Flip the coin until it comes up heads three times in a row. Now, if the man's theory is correct, it should be more likely that the next toss is tails. Toss the coin one more time and record the result.

Repeat this exercise as many times as it takes to convince you that after three heads in a row, the chances of getting heads are still $50-50$. It should only take you a few minutes.

Exercise #2:

Go to the bank and get a ten dollars in pennies. That's $1000$ pennies. (Or, whatever the cheapest coin is in the country where you live.) Again, get a piece of paper, and two big jars.

Put the pennies in the first big jar and shake them and dump them out. Take every penny that landed "heads" and put it back in the first jar; put the tails in the second jar. There should be about $500$ in each.

Now do it again, but only flip the ones that came up heads. Again, separate out the heads from the tails. Now there should be about $250$ coins that have come up heads twice, and $750$ that came up tails at least once.

Do it again. Keep on doing it until you either have no coins in the heads jar, or you have a group of coins that were flipped and came up heads nine times in a row. If you have none left, start over.

OK, you now have at least one coin that just came up heads nine times in a row. Flip it, and record the results. According to the man's theory, that coin should almost never be heads. What is it in reality?

Again, repeat the experiment until you have convincing evidence that coins do not remember what happened to them in the past.

Tags:

Normal Distribution

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