Goals
Notes
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Conditional Probability
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Independent and Dependent Events
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The Multiplication Rule
Conditional Probability
Notes
The table below gives the results of a study on children’s IQ and the
presence of a specific gene in the child.
High IQ
Normal IQ
Total
Gene
Present
33
39
72
Gene
Not Present
19
11
30
Total
52
50
102
A child is selected at random.
What is the probability that a child has a high IQ?
What is the probability that a child has a high IQ given that the child
has the gene?
Conditional Probability
Conditional probability of an event B given that an event A has
occured is denoted
P(B|A)
Ex Two cards are selected in sequence from a standard deck. Find
the probability that the second card is a queen, given that the first
card is a king (assume this first card is not replaced).
Ex: In the IQ and gene study, find the probability that a child does
not have the gene.
Then find the probability that a child does not have the gene, given
that the child has a normal IQ.
Can you infer from the data that the presence of the gene affects a
child’s IQ?
Notes
Independence
Notes
Consider selecting two cards from a standard deck, but replacing the
first card after you select it.
Ex: Find the probability that the second card is a queen, given that
the first card is a king (assuming that the first card is replaced.)
What inference can you make?
Ex: Can the events A and B affect each other?
1) (A) Exercising Frequently and (B) having a 4.0 GPA
2) (A) Driving over 85 miles per hour and (B) getting in a car
accident.
Independence
Notes
Two events are independent if the occurance of one of the events
does not affect the probability of the occurance of another event.
Mathematically,
P(B|A) = P(B) if A occurs.
Ex: Flip a coin twice. Are the events A: ”first flip is heads” and
B: ”second flip is tails” independent events?
Probability Experiment: ”Flip a coin twice.”
Sample space: {HH, HT , TH, TT }.
Events: A = ”First flip is heads, {HH, HT }”,
B = ”Second flip is tails, {HT , TT }”
Note that P(A) =
2
4
and P(B) = 24 .
The Multiplication Rule
We see from the last example that
1
P(A and B) = .
4
Notice that (1) A and B are independent events and (2)
P(A) · P(B) = (0.5)(0.5) = (0.25).
We can summarize this as:
The Multiplication Rule for Independent Events
If two events A and B are independent, then numerically,
P(A and B) = P(A) · P(B)
P(A and B) is sometimes called a joint probability.
Notes
The Multiplication Rule
Notes
That rule is special because we use it when two event are
independent. Generally, the following is true:
The General Multiplication Rule
If two events A and B occur in sequence, then numerically,
P(A and B) = P(A) · P(B|A)
Ex: Two cards are selected from a standard deck without
replacement. Find the probability that they are both hearts.
Examples
Notes
Ex: The probability that a rotary cuff surgery is successful is 0.9.
Assume that five different patients have the same rotary cuff surgery.
Find:
(a) The probability that five surgeries are successful.
(b) The probability that none of the five surgeries are successful.
(c) The probability that at least one of the five surgeries is successful.
Ex: In a jury selection pool, 65% of the people are female. Of these
65%, one out of four works in a health field. Find:
(a) The probability that a randomly selected person from the jury
pool is female and works in a health field.
(b) The probability that a randomly selected person from the jury
pool is female and does not work in a health field.
Assignment
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§3.2: #2, #19, #22, #31, #39, #40, #41
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Read 3.3.
Suggested Exercises: #1, #8, #10, #12, #17, #21
Notes