MATH 180A Lecture 6

Section 2.2

Bayes’ Rule

A flu test is 99% accurate, meaning that the false positive rate is 1% and the false negative rate is 1%. Currently, 0.33% of the US population has the flu. A randomly chosen person from the US population tests positive for the flu. What is the probability that this person actually has the flu, roundest to the nearest percent?

a. 99%
b. 75%
c. 25%
d. 0.33%
F = “person has the flu” 1st stage event
P = “person tests positive” 2nd stage event

Example. 90% of coins are fair. 9% of coins are slightly biased, meaning they land heads 60% of the time, and 1% of coins are heavily biased, meaning they land heads 80% of the time. I pick up a random coin off the street and flip it. It lands heads. What is the probability that the coin I picked up is heavily biased?

Asking for
Use Bayes’ Rule

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Bayes' Rule

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Fair: , Lightly Biased: , Heavily Biased:

Section 2.3

Independence

I flip a fair coin 2 times. What is the probability that the 2nd flip is heads? Suppose I told you that the 1st flip is tails. Now what is the probability that the 2nd flip is heads?

Independent

Two events in a probability space are independent if any of the three equivalent conditions hold:

Why are these equivalent?

Example. A bag contains 4 red marbles and 7 blue ones. Two are sampled. Let be the event that the 1st marble is red and the event that the 2nd marble is blue.

If the sampling is done w/ replacement, are $A,B$ independent?

independent

If the sampling is done w/o replacement, are $A,B$ independent?

not independent