Question: Independence vs Conditional Independence There are 3 coins. One is a fair coin, the second one is a biased coin with probability of landing heads

Independence vs Conditional Independence

There are 3 coins. One is a fair coin, the second one is a biased coin with probability of landing heads 0.8 and the third one is a biased coin with probability of landing heads 0.3. Someone hands you over one of these coins (selected randomly with equal probability).

(a) What is the probability of getting heads in your first toss?

(b) Given that you got heads in the first toss, what is the probability that you will also get a head in the second toss (assume that, given the coin, the outcomes of the tosses are independent)?

(c) Explain the distinction between assuming that the outcomes of the tosses are in- dependent and assuming that they are conditionally independent given the choice of the coin. Which of these assumptions seems more reasonable, and why?

(Hint: Imagine tossing the coin a 100 times and you get 80 heads. What do you think is the probability of heads in the 101st toss given this information? Now imagine that instead of 80 heads you got 50 heads in the first 100 times. What do you think is the probability of heads in the 101st toss given this information? Does your answer change? If the tosses were actually independent would your answers change? You can replace 100 with any huge number and keep the proportion of heads the same if that helps you.)

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