The probability that a coin has to land Heads is unknown, say p, where p is some number in [0,1]. Let X be a random variable that takes the value 1 if the coin lands Heads and 0 if the coin lands Tails.
(a) What is the probability distribution of X? What is its expected value and standard deviation?
(b) Suppose that if the coin lands Heads, Tom must give John $10, whereas if the coin lands Tails, John must give Tom $c, where c is some positive number. What is the random variable that expresses John's profit? What is the value of c that makes this game "fair"?
(c) Suppose that if the coin lands Heads, Kate must give Mary $1, whereas if the coin lands Tails, there will be no exchange of money. What is the random variable that expresses Mary's profit? How much money should Mary give to Kate (before the coin is tossed) in order to convince her to play this game?