Suppose Emi has 2 coins: one is a typical fair coin, the other is rigged and has "Heads" on both sides. Emi picks one at random (50% chance of picking each) and is curious to see which one she got. But there's a catch: Emi is not allowed to turn the coin over in her hand. She can only move the coin by tossing it in the air. After the coin lands, she can look to see what it shows.

a. Suppose Emi tosses the coin once, and sees that the result is Heads. What are the odds that she has the rigged coin? What about if the result is "Tails"?

b. Now suppose Emi repeats this action 8 times, and each time the result is "Heads". Does Emi definitely have the rigged coin? If not, find the probability that this is the case.

c. Consider again the question from part b, but this time Emi picked the coin from an urn of 1,000 coins, which contained 999 regular coins and 1 Heads/Heads coin: if she threw the coin 8 times and the result was Heads each time, what is the probability she has the rigged coin?

Please show the detail of each part