Discrrete mathematics for computer science conditional

What is a conditional probability?• Now suppose we know that the first roll is 4 or 5... Conditional Probability and Independence• Fact: A and B are independent events iff PrA|B = PrA..

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,

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

Pr(W | B) = | Ω ∩ Β |

| Β | = |

Ω ∩ Β | / | Σ |

| Β | / | Σ | = Πρ(

Ω ∩ Β ) Πρ( Β )

Pr(A | B) = Πρ( Α ∩ Β )

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|

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

A∩B

• 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:

Pr(F | W ) = Πρ( Ω ∩ Φ )

Πρ( Ω ) = 5 / 42 17 / 42 = 5 17

Conditional Probability and Independence

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

• That is, knowing whether B is the case gives no information that would help determine the probability of A

• Proof:

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:

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

_

A

S

“Total probability” = weighted average of

probabilities

• Pr(Santorum|Brokered) = 65

• Pr(Santorum|¬Brokered) = 05

• Pr(Brokered) = 07

• Then Pr(Santorum) =

65∙.07 + 05∙.93 = 092

FINIS