3.2 Independent and mutually exclusive events  (Page 2/23)

Flip two fair coins. (This is an experiment.)

The sample space is { HH , HT , TH , TT } where T = tails and H = heads. The outcomes are HH , HT , TH , and TT . The outcomes HT and TH are different. The HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads.

• Let A = the event of getting at most one tail . (At most one tail means zero or one tail.) Then A can be written as { HH , HT , TH }. The outcome HH shows zero tails. HT and TH each show one tail.
• Let B = the event of getting all tails. B can be written as { TT }. B is the complement of A , so B = A′ . Also, P ( A ) + P ( B ) = P ( A ) + P ( A′ ) = 1.
• The probabilities for A and for B are P ( A ) = $\frac{3}{4}$ and P ( B ) = $\frac{1}{4}$ .
• Let C = the event of getting all heads. C = { HH }. Since B = { TT }, P ( B AND C ) = 0. B and C are mutually exclusive. ( B and C have no members in common because you cannot have all tails and all heads at the same time.)
• Let D = event of getting more than one tail. D = { TT }. P ( D ) = $\frac{1}{4}$
• Let E = event of getting a head on the first roll. (This implies you can get either a head or tail on the second roll.) E = { HT , HH }. P ( E ) = $\frac{2}{4}$
• Find the probability of getting at least one (one or two) tail in two flips. Let F = event of getting at least one tail in two flips. F = { HT , TH , TT }. P ( F ) = $\frac{3}{4}$