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…