Discrrete mathematics for computer science conditional

Conditional Probability


What is a conditional probability?

•
•
•
•
•

It is the probability of an event in a subset of the sample space
Example: Roll a die twice, win if total ≥ 9
Sample space S = set of outcomes
= {11, 12, 13, 14, 15, 16, 21, 22, …, 65, 66}

Event W = pairs that sum to ≥ 9
= {36, 45, 46, 54, 55, 56, 63, 64, 65, 66}

Pr(W) = 10/36


What is a conditional probability?

•
•
•
•
•

Now suppose we know that the first roll is 4 or 5. What is now the

probability that the sum of the two rolls will be ≥ 9?
Let B = first roll is 4 or 5
= {41, 42, …, 46, 51, 52, …, 56}
Event W∩B = {45, 46, 54, 55, 56}
Pr(W | B) = |W∩B|/|B| = 5/12
“Probability of W given B”


Conditional probability

•

But since the sample space is the same,

| Ω ∩ Β | | Ω ∩ Β | / | Σ | Πρ(Ω ∩ Β)
Pr(W | B) =
=
=
| Β|
| Β| / | Σ|
Πρ( Β)

•

In general, the conditional probability of event A given event B is
defined as

Πρ( Α ∩ Β)
Pr(A | B) =
Πρ( Β)

What is the difference between
Pr(A|B) and Pr(B|A)?

•

Pr(A|B) is the proportion of B that is also within A, that is, Pr(A|B) is |
A∩B| as a proportion of |B|

A

•

B

Pr(A|B) is close to 1 but Pr(B|A) is close to 0

A∩B


CS20

•
•
•

This class has 42 students, 13 freshmen, 17 women, and 5 women
freshmen
So if a student is selected at random,

– Pr(Freshman) = 13/42,
– Pr(Woman) = 17/42
– Pr(Woman freshman) = 5/42.

If a random selection chooses a woman, what is the probability she is
a freshman?

– Simple way: #women freshmen/#women = 5/17
– Using probability:

Πρ(Ω ∩ Φ) 5 / 42
5
Pr(F | W ) =
=
=
Πρ(Ω )
17 / 42 17


Conditional Probability and Independence

•
•

Fact: A and B are independent events iff Pr(A|B) = Pr(A).

•

Proof:

That is, knowing whether B is the case gives no information that
would help determine the probability of A.
A and B independent iff Pr(A)∙Pr(B) = Pr(A∩B)
Pr(A∩B) = Pr(A|B)∙Pr(B)
So as long as Pr(B) is nonzero,
Pr(A)∙Pr(B) = Pr(A|B)∙Pr(B) iff Pr(A) = Pr(A|B)


Total Probability

•

Suppose (hypothetically!):

– Rick Santorum has a 5% probability of getting enough delegates to

become the Republican nominee, unless the voting goes beyond the
first ballot and there is a brokered convention

– In a brokered convention, Santorum has a 65% probability of winning
the nomination

•

– There is a 7% probability of a brokered convention (cf. Intrade.com)
What is the probability that Santorum will be the Republican
nominee?

Total Probability
Simple version: For any events A and B whose probability is neither
0 nor 1:

Pr(A) = Πρ( Α | Β)⋅ Πρ( Β) + Πρ( Α | Β)⋅ Πρ( Β)
That is, Pr(A) is the weighted average of the probability of A
conditional on B happening, and the probability of A conditional
on B not happening.

_
B

B
A
S


“Total probability” = weighted average of
probabilities

Pr(S) = Πρ(Σ | Β)⋅ Πρ( Β) + Πρ(Σ | Β)⋅ Πρ( Β)

•
•
•
•

Pr(Santorum|Brokered) = .65
Pr(Santorum|¬Brokered) = .05
Pr(Brokered) = .07

Then Pr(Santorum) =

.65∙.07 + .05∙.93 = .092


FINIS