Measure, Integration & Real Analysis, 2021a

384 Chapter 12 Probability Measures
12.8 Example independent events: coin tossing
Suppose Ω = {H, T} 4 , where H and T are symbols that you can think of as
denoting “heads” and “tails”. Thus elements of Ω are 4-tuples of the form
ω =(ω 1 , ω 2 , ω 3 , ω 4 ),
where each ω j is H or T. Let F be the collection of all subsets of Ω, and let
P =(counting measure on Ω)/16, as we expect from a fair coin toss.
Let
A = {ω ∈ Ω : ω 1 = ω 2 = ω 3 = H} and B = {ω ∈ Ω : ω 4 = H}.
Then A contains two elements and thus P(A) = 1 8 , corresponding to probability 1 8
that the first three coin tosses are all heads. Also, B contains eight elements and thus
P(B) = 1 2 , corresponding to probability 1 2
that the fourth coin toss is heads.
Now
P(A ∩ B) =
16 1 = P(A) · P(B),
where the first equality holds because A ∩ B consists of only the one element
(H, H, H, H) and the second equality holds because P(A) = 1 8 and P(B) = 1 2 .
The equation above shows that A and B are independent events.
If we toss a fair coin many times, we expect that about half the time it will be
heads. Thus some people mistakenly believe that if the first three tosses of a fair
coin are heads, then the fourth toss should have a higher probability of being tails,
to balance out the previous heads. However, the coin cannot remember that it had
three heads in a row, and thus the fourth coin toss has probability 1 2
of being heads
regardless of the results of the three previous coin tosses. The independence of the
events A and B above captures the notion that the results of a fair coin toss do not
depend upon previous results.
12.9 Example independent events: product probability space
Suppose (Ω 1 , F 1 , P 1 ) and (Ω 2 , F 2 , P 2 ) are probability spaces. Then
(Ω 1 × Ω 2 , F 1 ⊗F 2 , P 1 × P 2 ),
as defined in Chapter 5, is also a probability space.
If A ∈F 1 and B ∈F 2 , then (A × Ω 2 ) ∩ (Ω 1 × B) =A × B. Thus
(P 1 × P 2 ) ( (A × Ω 2 ) ∩ (Ω 1 × B) ) =(P 1 × P 2 )(A × B)
= P 1 (A) · P 2 (B)
=(P 1 × P 2 )(A × Ω 2 ) · (P 1 × P 2 )(Ω 1 × B),
where the second equality follows from the definition of the product measure, and
the third equality holds because of the definition of the product measure and because
P 1 and P 2 are probability measures.
The equation above shows that the events A × Ω 2 and Ω 1 × B are independent
events in F 1 ⊗F 2 .
Measure, Integration & Real Analysis, by Sheldon Axler