Conditional Probability: What

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Section 5.3

• Conditional Probability Whats the Probability
  of A, Given B?

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Conditional Probability

• For events A and B, the conditional
• probability of event A, given that event B has
  occurred is

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Conditional Probability
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Example What are the Chances of a Taxpayer
being Audited?
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Example Probabilities of a Taxpayer Being
Audited
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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 Taxpayers income 100,000

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Example Probabilities of a Taxpayer Being
Audited
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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

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Example The Triple Blood Test for Down Syndrome

• 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

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Example The Triple Blood Test for Down Syndrome

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

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Example The Triple Blood Test for Down Syndrome
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Example The Triple Blood Test for Down Syndrome

• 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

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Example The Triple Blood Test for Down Syndrome

• P(POS) 1355/5282 0.257

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Example The Triple Blood Test for Down Syndrome

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

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Example The Triple Blood Test for Down Syndrome
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Example The Triple Blood Test for Down Syndrome

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

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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(AB) x P(B)
• also
• P(A and B) P(BA) x P(A)

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Example How Likely is a Double Fault in Tennis?

• Roger Federer 2004 mens 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

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Example How Likely is a Double Fault in Tennis?

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

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Example How Likely is a Double Fault in Tennis?

• P(F1) 0.36
• P(F2F1) 0.06
• P(F1 and F2) P(F2F1) x P(F1)
• 0.06 x 0.36 0.02

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Sampling Without Replacement

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

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Example How Likely Are You to Win the Lotto?

• In Georgias 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?

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Example How Likely Are You to Win the Lotto?

• P(have all 6 numbers) P(have 1st and 2nd and
  3rd and 4th and 5th and 6th)
• P(have 1st)xP(have 2ndhave 1st)xP(have 3rd
  have 1st and 2nd) P(have 6thhave 1st, 2nd, 3rd,
  4th, 5th)

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Example How Likely Are You to Win the Lotto?

• 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44
• 0.00000007

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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

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Independent Events Defined Using Conditional
Probabilities

• Events A and B are independent if
• P(AB) P(A)
• If this holds, then also P(BA) P(B)
• Also, P(A and B) P(A) x P(B)

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Checking for Independence

• Here are three ways to check whether events A and
  B are independent
• Is P(AB) P(A)?
• Is P(BA) 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

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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

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Example How to Check Whether Two Events are
Independent

• 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
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Example How to Check Whether Two Events are
Independent

• Are the events POS and D independent or
  dependent?
• Is P(POSD) P(POS)?

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Example How to Check Whether Two Events are
Independent

• Is P(POSD) P(POS)?
• P(POSD) P(POS and D)/P(D)
• 0.009/0.010 0.90
• P(POS) 0.256
• The events POS and D are dependent

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Section 5.4

• Applying the Probability Rules

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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

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Example Is a Matching Birthday Surprising?

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

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Example Is a Matching Birthday Surprising?

• P(at least one match) 1 P(no matches)

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Example Is a Matching Birthday Surprising?

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

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Example Is a Matching Birthday Surprising?

• P(no matches)
• (365/365) x (364/365) x (363/365) x
• x (341/365)
• P(no matches) 0.43

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Example Is a Matching Birthday Surprising?

• P(at least one match)
• 1 P(no matches) 1 0.43 0.57