Section 5.3

Agresti/Franklin Statistics, 1 of 87

Section 5.3

Conditional Probability: What’s the Probability of A, Given B?


Conditional Probability

For events A and B, the conditional

probability of event A, given that event B has occurred is:

)(

) ()|P(

BP

BandAPBA


Conditional Probability


Example: What are the Chances of a Taxpayer being Audited?


Example: Probabilities of a Taxpayer Being Audited



What was the probability of being audited, given that the income was ≥ $100,000?

• Event A: Taxpayer is audited

• Event B: Taxpayer’s income ≥ $100,000



007.01334.0

0010.0

P(B)

B) andP(A B)|P(A


Example: The Triple Blood Test for Down Syndrome

A positive test result states that the condition is present

A negative test result states that the condition is not present



False Positive: Test states the condition is present, but it is actually absent

False Negative: Test states the condition is absent, but it is actually present



A study of 5282 women aged 35 or over analyzed the Triple Blood Test to test its accuracy





Assuming the sample is representative of the population, find the estimated probability of a positive test for a randomly chosen pregnant woman 35 years or older



P(POS) = 1355/5282 = 0.257


Given that the diagnostic test result is positive, find the estimated probability that Down syndrome truly is present




035.0257.0

009.0

5282/1355

5282/48

P(POS)

POS) and P(D POS)|P(D


Summary: Of the women who tested positive, fewer than 4% actually had fetuses with Down syndrome



Multiplication Rule for Finding P(A and B)

For events A and B, the probability that A and B both occur equals:

• P(A and B) = P(A|B) x P(B)

also

• P(A and B) = P(B|A) x P(A)


Example: How Likely is a Double Fault in Tennis?

Roger Federer – 2004 men’s champion in the Wimbledon tennis tournament• He made 64% of his first serves

• He faulted on the first serve 36% of the time

• Given that he made a fault with his first serve, he made a fault on his second serve only 6% of the time



Assuming these are typical of his serving performance, when he serves, what is the probability that he makes a double fault?



P(F1) = 0.36 P(F2|F1) = 0.06

P(F1 and F2) = P(F2|F1) x P(F1)

= 0.06 x 0.36 = 0.02


Sampling Without Replacement

Once subjects are selected from a population, they are not eligible to be selected again


Example: How Likely Are You to Win the Lotto?

In Georgia’s Lotto, 6 numbers are randomly sampled without replacement from the integers 1 to 49

You buy a Lotto ticket. What is the probability that it is the winning ticket?



P(have all 6 numbers) = P(have 1st and 2nd and 3rd and 4th and 5th and 6th)

= P(have 1st)xP(have 2nd|have 1st)xP(have 3rd| have 1st and 2nd) …P(have 6th|have 1st, 2nd, 3rd, 4th, 5th)



6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44

= 0.00000007


Independent Events Defined Using Conditional Probabilities

Two events A and B are independent if the probability that one occurs is not affected by whether or not the other event occurs


Independent Events Defined Using Conditional Probabilities

Events A and B are independent if:

P(A|B) = P(A)

If this holds, then also P(B|A) = P(B)

Also, P(A and B) = P(A) x P(B)


Checking for Independence

Here are three ways to check whether events A and B are independent:

• Is P(A|B) = P(A)?

• Is P(B|A) = P(B)?

• Is P(A and B) = P(A) x P(B)?

If any of these is true, the others are also true and the events A and B are independent


Example: How to Check Whether Two Events are Independent

The diagnostic blood test for Down syndrome:

POS = positive result

NEG = negative result

D = Down Syndrome

DC = Unaffected



Blood Test:

Status POS NEG Total

D 0.009 0.001 0.010

Dc 0.247 0.742 0.990

Total 0.257 0.743 1.000


Are the events POS and D independent or dependent?

• Is P(POS|D) = P(POS)?




Is P(POS|D) = P(POS)?

P(POS|D) =P(POS and D)/P(D)

= 0.009/0.010 = 0.90

P(POS) = 0.256

The events POS and D are dependent


Section 5.4

Applying the Probability Rules


Is a “Coincidence” Truly an Unusual Event?

The law of very large numbers states that if something has a very large number of opportunities to happen, occasionally it will happen, even if it seems highly unusual


What is the probability that at least two students in a group of 25 students have the same birthday?

Example: Is a Matching Birthday Surprising?


P(at least one match) = 1 – P(no matches)



P(no matches) = P(students 1 and 2 and 3 …and 25 have different birthdays)




P(no matches) =

(365/365) x (364/365) x (363/365) x …

x (341/365)

P(no matches) = 0.43


P(at least one match) =

1 – P(no matches) = 1 – 0.43 = 0.57


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Section 5.3

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