Download Lectures 3 and 4 - UCSD Math Department

Lectures 3 and 4
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2.1 Two consequences of the axioms
One of the main reasons for introducing sets at the beginning was to determine formulas
for the probability of unions, intersections, and complements of sets, using the axioms
of probability. These general rules apply in any probability space. Throughout this
section, (Ω, F, P ) is a probability space. The first consequence of the axioms is the rule
of complements. Recall the complement of a set A, which is denoted A1 is
Ω\A = {ω ∈ Ω : ω 6∈ A}.
This is the set of outcomes which is not in A. From the axioms, we obtain:
Proposition 1 Let A be an event. Then
P (A) = 1 − P (A).
Proof . The events A and A partition Ω, by definition, so from Axioms 1 and 3,
1 = P (Ω) = P (A) + P (A).
This gives the formula.
The above formula is useful, as it is sometimes much easier to compute P (A) than
P (A), or vice versa. The inclusion-exclusion formula is a general rule which helps to
compute combinatorial probabilities, and follows as a useful generalization of Axiom 3
of probability.
Proposition 2 (Inclusion-Exclusion Formula) Let A and B be events in a probability space. Then
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
1
Some books write Ac instead
1
Proof . We may write A ∪ B as a disjoint union of the following three sets:
A ∪ B = (A\B) t (B\A) t (A ∩ B)
So by Axiom 3 of probability,
P (A ∪ B) = P (A\B) + P (B\A) + P (A ∩ B).
Now A\B and A ∩ B are two disjoint sets whose union is A. So using Axiom 3,
P (A) = P (A\B) + P (A ∩ B).
This gives P (A\B) = P (A)−P (A∩B). Similarly, P (B\A) = P (B)−P (A∩B). Putting
these two equations into the equation for P (A ∪ B), we get
P (A ∪ B) = P (A) − P (A ∩ B) + P (B) − P (A ∩ B) + P (A ∩ B)
= P (A) + P (B) − P (A ∩ B).
This completes the proof.
We could actually take unions of even more sets, for example, for three sets one obtains
an inclusion-exclusion formula for three sets:
P (A∪B ∪ C) = P (A)+P (B)+P (C)−P (A ∩ B)−P (A ∩ C)−P (B ∩ C)+P (A∩B ∩ C).
2.2 Computing probabilities.
Let’s do examples of how to use the formulas for sets and probabilities to compute a
probability.
First Example: Fair dice. Two dice are tossed, and the sum of the two numbers is
recorded. The sample space Ω is the set of all ordered pairs of numbers up to six, i.e.
Ω = {(1, 1), (1, 2), (2, 1), . . . , (6, 6)}.
The events are all the subsets of Ω, and the probability measure we will use is
P (A) =
2
|A|
.
36
Note that we could imagine this as the probability measure which comes from tossing
two fair dice independently of one another. It is also the uniform measure on Ω, since
P ({(i, j)}) = 1/36 for every pair (i, j) ∈ Ω – this means that i was tossed on the first
die, and j was tossed on the second. Consider now the event A that the sum of the two
numbers on the dice is a single digit number, namely 1, 2, 3, 4, 5, 6, 7, 8 or 9. We could
directly write down all the outcomes in A, there are 30 of them, and then we could
compute P (A) = |A|/36 = 30/36 = 5/6. But it is much easier to use the complement
rule: A is the event that the sum is 10, 11 or 12, so
A = {(4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6)}
and then
P (A) =
|A|
6
1
=
= .
36
36
6
By the complement rule,
P (A) = 1 − P (A) =
5
6
which agrees with the preceding answer.
Second Example: Two fair dice. If two fair dice are thrown, what is the chance
that at least one of the dice shows an odd number? Let A be the event that the first die
shows an odd number, and let B be the event that the second shows an odd number. We
are looking for P (A ∪ B) in the probability space (Ω, F, P ) given in the first example.
Now we know by the inclusion-exclusion formula that
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
We carefully write down A:
A = {(1, 1), (1, 2), . . . , (1, 6), (3, 1), (3, 2), . . . , (3, 6), (5, 1), (5, 2), . . . , (5, 6)}.
There are 18 outcomes in A, so
P (A) =
|A|
1
= .
36
2
We could have argued this in a simpler way: half the numbers on the first die are odd,
so there is a fifty percent chance that the first die shows an odd number, regardless of
what the second die shows. Similarly, we get P (B) = 1/2. Now A ∩ B is the event that
both numbers are odd, which is 1/4, since
A ∩ B = {(1, 1), (3, 3), (5, 5), (1, 3), (3, 1), (1, 5), (5, 1), (3, 5), (5, 3)}
3
and so
9
1
|A ∩ B|
=
= .
36
36
4
Again we could have argued more simply: there are three odd numbers possible on the
first die, and three on the second, so there are nine combinations which give two odd
numbers. Finally, by inclusion-exclusion,
P (A ∩ B) =
P (A ∪ B) =
1 1 1
3
+ − = .
2 2 4
4
We remark that whenever one sees in a probability problem the phrase at least one, this
indicates the union of some events.
The purpose of the next example is to illustrate how one has sometimes to use careful
logic to compute the required probability in an efficient way.
Third Example: Logic and sets. Two fair six-sided dice are thrown, with probability
measure P (A) = |A|/36 in the same probability space as the last two examples. Let
A be the event that the sum of the dice is odd, or at least one of the numbers is odd
and both dice show different numbers. To compute P (A), we could list all outcomes
(as an exercise) in A and we would find that there are 24 pairs of numbers in A, so
P (A) = 24/36 = 2/3. However,
A = B ∪ (C ∩ D)
where B is the event that the sum of the dice is odd, C is the event that at least one of
the numbers is odd, and D is the event that both dice show different numbers. Therefore
P (A) = P (B ∪ (C ∩ D)) = P (B) + P (C ∩ D) − P (B ∩ C ∩ D)
using inclusion-exclusion. Observe that B ∩ C ∩ D = B, for if B occurs (sum is odd),
then C must occur, and D must also occur. Therefore
P (A) = P (C ∩ D).
Now C ∩ D is the event that one of the dice shows an odd number and the other shows
a different number. It is easier to deal with C ∩ D = C ∪ D by deMorgan’s law – this
is the event that both dice show an even number or both dice show the same number.
Then by inclusion-exclusion
P (C ∩ D) = P (C ∪ D) = P (C) + P (D) − P (C ∩ D).
4
Now P (C) = 1/4 (probability both dice are even) and P (D) = 1/6 (probability both
dice show the same number) and C ∩ D = {(2, 2), (4, 4), (6, 6)} (both dice are even and
show the same number). Therefore
P (C ∩ D) =
1 1
3
1
+ −
= .
4 6 36
3
By the complement rule,
P (A) = P (C ∩ D) = 1 − P (C ∩ D) = 1 −
2
2
= .
3
3
This answers the problem, but it is not the only way to do this problem. It is important
to become familiar with manipulating sets to compute probabilities.
Fourth Example: Probability in the integers. Let Ω = Z and let F consist of all
subsets of Ω. Suppose we define
1
P ({ω}) = ω .
2
Then P defines a probability measure by letting for any A ⊂ Ω
X
P (A) =
2−ω .
ω∈A
To see this, certainly P (A) ≥ 0 for every A and
P (Ω) =
X
2−ω =
ω∈Z
1 1 1
+ + + ··· = 1
2 4 8
since the sum is a geometric series. In general, if P is a probability measure defined on
the integers, then the requirement
X
P ({ω}) = 1
ω∈Z
implies from elementary calculus that
lim P ({ω}) = 0
ω→∞
in other words, larger and larger integers must have smaller and smaller probabilities.
Fifth Example: Combinatorial probability. We are going to spend some time later
on looking at combinatorial probability, here we do a first example. Consider a standard
deck of 52 cards (thirteen cards in each of four suits, diamonds, clubs, spades and hearts,
and thirteen different card values 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, four of each value –
5
for example the diamonds and spades are shown below). Suppose we consider drawing
five cards from the deck with the uniform probability measure on all five card hands.
Thus the sample space Ω is the set of all five card hands, F is the set of all subsets of
Ω and, for any event A, P (A) = |A|/|Ω|. Here Ω is very large:
|Ω| = 52 · 51 · 50 · 49 · 48
is the total number of five card hands. Let’s compute the probability that at least two of
the cards in a five card hand are of the same value (in poker, this is called a pair). This
is a counting problem: how many ways can we fill five slots with five cards such that two
of the cards are of the same suit (this is |A|), and then we divide by |Ω| = 52 · 51 · · · 48 to
get the probability. It turns out to be easier to compute P (A) – this is the probability
that all cards have different values. There are 52 choices to fill the first slot, but now
the second slot can’t have a card of the same value, so there are only 48 choices for the
second slot. Similarly there are 44 choices for the third, 40 for the fourth, and 36 for
the last. So
52 · 48 · 44 · 40 · 36
P (A) =
= 0.50708 . . .
52 · 51 · 50 · 49 · 48
Therefore
P (A) = 0.49292 . . .
So it is in fact more likely than not that all the cards will be different (but perhaps
surprisingly close to 1/2).
Figure 1 : Diamonds and spades
jpg
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