(a) The man shuts his eyes, chooses a coin at random, and tosses it. What is the
ID: 2964350 • Letter: #
Question
(a) The man shuts his eyes, chooses a coin at random, and tosses it. What is the
probability that the lower face of the coin is heads?
(b) He opens his eyes and sees that the upper face of the coin is a head. What is the
probability that the lower face is a head.
(c) He shuts his eyes again, picks up the same coin, and tosses it again. What is the
probability that the lower face is a head?
(d) He opens his eyes and sees that the upper face is a head. What is the probability
that the lower face is a head?
Explanation / Answer
a) Note that *two*, not one, of the coins are normal; the total number of coins is 5, not 4.
So P(H on lower face) should be (1/5)(1+1+0+1/2+1/2) = 3/5.
b) P(H on lower face given H on upper face)
= P(H on both faces)/P(H on upper face)
= P(H on both faces)/P(H on lower face)
= (2/5)/(3/5)
= 2/3, so you answered this part correctly (good job!).
c) From b), we know that, given that the upper face is a head after the first toss, the coin is a two-headed coin with probability 2/3, a two-tailed coin with probability 0, and a normal coin with probability 1/3.
So the probability that this same coin, when tossed again, shows heads on the lower face is (2/3)(1) + (0)(0) + (1/3)(1/2) = 5/6.
d) From b), we know that, given that the upper face is a head after the first toss, the coin is a two-headed coin with probability 2/3, a two-tailed coin with probability 0, and a normal coin with probability 1/3.
In this situation, we know from c) that the probability that this same coin, when tossed again, shows heads on the lower face is 5/6. Therefore, the probability that this same coin, when tossed again, shows heads on the upper face is also 5/6.
Also in this situation, we already know that the probability that this same coin, when tossed again, shows heads on both faces is 2/3.
Therefore we conclude that, given that the upper face is a head the second time around, the probability that the lower face is a head is (2/3)/(5/6) = 4/5.
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