Conditional Probability

There are three fair coins. Two of them are normal coins and the remainig one is a two-head coin. A coin is chosen at random and tossed.
Q: If a head shows up, this chosen coin is tossed once more. If a tail shows up, another coin is chosen from the remaining two coins and tossed. Find the probability of getting
(i) two heads in these two trails
第一次取後雙面都是公的硬幣概率 = 1/3
則擲到兩次公的概率 = 1
所以這種況取後兩公的概率是 (1/3)(1) = 1/3
第一次取得普通硬幣的概率 = 2/3
則擲到兩次公的概率 = 1/4
所以這種況取後兩公的概率是 = (2/3)(1/4) = 1/6
所以機會是 (1/3) + (1/6)
= 1/2
所以擲後兩公的概率是 1/2

(ii) two tails in these two trails
第一次取得普通硬幣的概率 = 2/3
第一次試擲結果是字的概率是 = 1/2
再取得普通硬幣的概率 = 1/2
第二次試擲結果是字的概率是 = 1/2
所以擲得兩次都是字的概率是
= (2/3)(1/2)(1/2)(1/2)
= 1/12
所以擲出兩字的概率是 1/12