Bayes inference - a biased coin
Can you tell if a coin is fair or biased by tossing it multiple times? Will your assessment change the more times you toss? 1. You start with two hypoteses: H1 is the hypotesis that the coin is fair, and H2 is the hypotesis that it's biased (or unfair) - in our case - that it has 2 heads (no tails) 2. You start with a (prior) belief (probability) that the coin is biased (P(H2)) of for example, .25 (and therefore that the coin is fair (P(H1)) of .75 (1 - P(H2)) 3. If you toss the coin once and it comes up heads, will you change your belief in the hypothesis that this is a biased/unfair coin (H2)? 4. From the graph you can see that after 1 toss which came heads (h), your belief that the coin is biased will change (that is, P(H2 | h) will go up to 0.4) 5. Now your prior belief in H2 is higher (it went from 0.25 to 0.4), and you are ready to toss the coin again 6. As you can see from the graph, after 5 tosses, you could be more than 90% certain that this is a biased coin (if you start with an initial belief of 0.25, prior to tossing the coin for the first time)
You can explore how you'd change your belief in H2 depending on what your initial belief is. For example: if you start with a belief that with a high probability the coin is biased (say, P(H2) = .9), how many tosses will it take to convince you with better than .99 probability that the coin is biased? Will you need more or less tosses to get convinced to the same degree, if you start with a belief that the probability of the coin being biased is only 0.1 (that is, that with high probability, the coin is fair)?