1. A coin was flipped 100 times, and 55 times it showed up heads. Do you think the coin is biased? What about 550 heads with 1000 flips?
Assume the coin is not biased (null hypothesis), the probability of showing head is p = 0.5.
For the first case: z = 100^0.5 * (55/100 - 0.5)/(0.5*(1-0.5))^0.5 = 1, we cannot reject the null hypothesis.
For the second case: z = 1000^0.5 * (55/100 - 0.5)/(0.5*(1-0.5))^0.5 = 3.3, we can reject the null hypothesis.
2. There is a fair coin and an unfair coin (both sides faces). You pick one at random, flip it five times, and observe five faces. What is the probability that you pick the unfair coin?
P(fair) = P(unfair) = 0.5
P(unfair | event) = P(event | unfair) P (unfair) / (P(event | unfair) P (unfair) + P(event | fair) P (fair))
= 1 * 0.5 / (1 * 0.5 + 1/32 * 0.5) = 32/33
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