normal. He e shu huts his eyes, picks face of the coin is a head? IS probability that the lower He opens his eyes and sees tha coin is showing heads; what is the probability that the lower face is a head? He shuts his eyes again, and to the coin again. What is the probability that the lower face is a head? He opens his eyes and sees that coin is showing heads; what is the probability that the lower face is a head? He discards this coin, picks anotat random, and tosses it. What is the probability that it shows heads? 4. What do you think of the following 'proof' by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB) = P(BW) = P(WB) = P(WW) = . We add a black ball, so 1. that P(BBB) = P(BBW) = P(BWB = P(BWW) = . Next we pick a ball at random; the chance that the ball is black is (using condition probabilities) 1. 1½ + ½ ½ + ½ + ½ ½ = . However, if 31 · 1/1 1/3 · 31 · 1 3. there is probability that a ball, chorandomly from three, is black, then there must be two black nally there was one black and one white ball in the urn. and one white, which is to say that 5. The Monty Hall problem: goa and cars. (a) Cruel fate has made you a contestant in a game show; you have to choose one of three doors. One conceals a new car, two conceal old goats. You choose, but your chosen door is not opened immediately. Instead, the presenter opens another door to reveal a goat, and he offers you the opportunity to change your choice to the third door (unopened and so far unchosen). Let p be the (conditional) probability that the third door conceals the car. The value of p depends on the presenter' otocol. Devise protocols to yield the values p = ½, p = . protocol such that p = a. Are you well advised to change Show that, for a = [1, 3], there ex your choice to the third door? 1 (b) In a variant of this question, the center is permitted to open the first door chosen, and to reward