## Probability of bias

Another probability riddle that I pondered recently.

We toss a coin and get a head. What’s the probability that the coin is biased towards heads?

Does this question make sense at all? The probability p of throwing a head is a random variable. If we know the distribution of p then our question can be answered.

But what distribution should we assume for a real physical coin? I have no idea. It’s definitely not uniform and even not normal.

I know how to compute the answer if the distribution is uniform. The answer is

**3/4**.But what is the right question to this answer? I don’t want to resort to urns and balls. Elections would make a nice example, but in real life the vote ratio distribution is hardly uniform. Coke vs Pepsi? Okay here is a really stupid one: If we observe one molecule of CO2 in the atmosphere heading downwards, then what’s the probability that the sky is falling? It’s 3/4! Oh, well.

A sack contains a large number of apples and oranges. A randomly pulled object is an orange. What’s the probability that the sack contains more oranges than apples? Answer: 3/4.

What if we pull 2 oranges in a row? Then the answer is 7/8.

For n oranges the answer is 1 – 1/2^(n+1).

How to derive this? Some other time…

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On the “Son born on a Tuesday” probability puzzle « Drawing Blanks08/30/2010 at 12:11 am