Download Chapter 1.3 – Conditional probability and independence

Chapter 1.3 – Conditional probability and independence
• Consider the dart board below, where |A| = 1 (area of region A), |B| = 2, |C| = 3, |M | = 4.
Define the sample space as S = A ∪ B ∪ C ∪ M and the board as D = A ∪ B ∪ C.
M
A
B
C
• Assuming all points in S are equally likely, compute the probabilities of A, B, C, and D.
• Now say you are allowed to throw the dart until you hit the board. What is the probability of
A given that you hit the board?
• This is the conditional probability P (A|D), read as “probability of A given D”.
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• If A and B are events in S with P (B) > 0, the conditional probability of A given B is
P (A|B) =
P (A ∩ B)
.
P (B)
• Back to the dart board example, what is P (A|D)?
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• Prove that P (B|B) = 1.
• Prove that P (A|B) = 0 if A and B are disjoint.
• Prove that P (A|B) = 0 if A = ∅.
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• Prove that P (·|B) is a valid probability measure on sample space B.
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• Given that your hand has all face cards, what is the probability you have a full house?
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• Monte Hall problem: Suppose you are on a game show, and you are given the choice of 3
doors. There is a car behind one door and goats behind the others. You pick door #1 (but it
is not opened). The host, who knows what’s behind the doors, then opens door #3 which has
a goat. Then he says, ”Do you want to switch to door #2?” What should you do?
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• Bayes’ Rule offers a relationship between P (A|B) and P (B|A).
• Example: Let A be the event that a patient has a disease and B be the event that the patient
tests positive. We know:
P (B|A) = probability of testing positive given you have the disease
(1)
= probability of a true positive = 0.99.
P (B|AC ) = probability of testing positive given you do not have the disease
= probability of a false positive = 0.03.
P (A) = probability of a randomly selected person having the disease = prevalence = 0.05.
• Now a patient tests positive. What is P (A|B), the probability they have the disease given a
positive test?
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• In general, if P (B) > 0, then Bayes’ rule is
P (A|B) =
P (B|A)P (A)
P (B|A)P (A)
=
.
P (B)
P (B|A)P (A) + P (B|AC )P (AC )
• Back to the patient, what is P (A|B)?
1. Does P (A|B) increase if the test is more accurate with P (B|A) = 0.999, P (B|AC ) = 0.001,
and P (A) = 0.05?
2. Does P (A|B) increase if prevalence decreases P (B|A) = 0.99, P (B|AC ) = 0.03, and
P (A) = 0.01?
3. Does P (A|B) increase if the test is a coin flip P (B|A) = 0.5, P (B|AC ) = 0.5, and P (A) =
0.05?
4. Is P (A|B) > P (A|B C )?
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• There is a more general expression if the events can have more than two outcomes.
• If A1 , A2 , ... partition S and P (B) > 0, then
P (Aj |B) =
P (B|Aj )P (Aj )
P (B|Aj )P (Aj )
= ∑∞
.
P (B)
j=1 P (B|Aj )P (Aj )
• Example: A1 = a person lives in Raleigh; A2 = a person lives in Chapel Hill; A3 = a person
lives in Durham; B = a person is a criminal.
• We know the population distribution: P (A1 ) = 0.5, P (A2 ) = 0.2, and P (A3 ) = 0.3.
• We know the crime rates: P (B|A1 ) = 0.01, P (B|A2 ) = 0.02, and P (B|A3 ) = 0.02.
1. What is the probability that a randomly selected person is a criminal?
2. Given we identify a criminal, what is the probability they are from Raleigh?
3. Given we identify a criminal, what is the probability they are from Chapel Hill?
4. Given we identify a criminal, what is the probability they are from Durham?
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• Two events are independent if they do not influence each other.
• If two events are not independent, they are dependent.
• Three equivalent definitions of independence:
1. A and B are independent if P (A) = P (A|B).
2. A and B are independent if P (B) = P (B|A).
3. A and B are independent if P (A ∩ B) = P (A)P (B).
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• In the testing example with P (A) = 0.05, P (B|A) = 0.99, and P (B|AC ) = 0.03, are
testing and disease status independent?
• Can you think of a situation where testing and disease status would be independent?
• In the Monte Hall problem, are the events ”switch” and ”win” independent?
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• Example You are dealt two cards without replacement. A = you get at least one heart, B =
you get two sevens. Are A and B independent?
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• A1 , A2 , ..., An are mutually independent if for any subset i1 , ..., ik ∈ {1, ..., n},
(k
)
k
∏
∩
P
Ail =
P (Ail ).
l=1
l=1
• In particular, we must have
1. P (Ai ∩ Aj ) = P (Ai )P (Aj ) for all i and j.
2. P (Ai ∩ Aj ∩ Al ) = P (Ai )P (Aj )P (Al ) for all i, j, and l.
3. P (A1 ∩ ... ∩ An ) = P (A1 ) · ... · P (An ).
• Example: We plant three mutually independent seeds. Each has survival probability 0.4.
What is the probability of at least one surviving?
• How many should we plant if we want the probability of at least one surviving to be 0.99?
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• If A and B are independent, then A and B C , AC and B, and AC and B C are all independent.
• If A, B, and C are mutually independent, then AC , B , and C are mutually independent, etc.
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