Download Probability - Part 2 Chapter 3, part 2

Probability - Part 2
Chapter 3, part 2
Mary Lindstrom
(Adapted from notes provided by Professor Bret Larget)
February 1, 2004
Statistics 371
Last modified: February 1, 2004
Conditional Probability and Probability
Trees
It common in biological probability problems for an event to
consist of the outcomes from a sequence of possibly dependent
chance occurrences.
In this case, a probability tree is a very useful device for guiding
the appropriate calculations.
We have already discussed definitions of probability and events.
The following example will illustrate the use of the Conditional
Probability Rule, and the Rule of Total Probability.
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Example
The following relative frequencies are known from review of
literature on the subject of strokes and high blood pressure in
the elderly.
• 1. Ten percent of people aged 70 will suffer a stroke within
five years;
• 2. Of those individuals who had their first stroke within five
years after turning 70, forty percent had high blood pressure
at age 70;
• 3. Of those individuals who did not have a stroke by age 75,
twenty percent had high blood pressure at age 70.
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Example
Two questions of interest are:
• What is the probability that a 70 year-old patient has high
blood pressure?
• What is the probability that a 70 year-old patient with high
blood pressure will have a stroke within five years?
To answer these questions, it is useful to construct a probability
tree.
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Example
First let’s define
S = {stroke before age 75}
Now we can write the statement “Ten percent of people aged
70 will suffer a stroke within five years” as
Pr{S} = 0.10
Which tells us right away that
Pr{S c} = 1 − Pr{S} = 0.90
Where the notation S c means the complement of the event S.
Since S is stroke, S c is no stroke.
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Rule of complements
This is the rule of complements:
Pr{E c} = 1 − Pr{E}
Where E is any event.
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Probability tree
We can draw a picture of what we know so far
Pr{S} = 0.10
Pr{S c} = 0.90
as:
0.1
0.9
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Sc
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Example (cont.)
“Of those individuals who had their first stroke within five years
after turning 70, forty percent had high blood pressure at age
70” becomes
Pr{H | S} = 0.40
Where
H = {high blood pressure at age 70}
The symbol | is read “given” and indicates that 0.40 is a
conditional probability.
“Of those individuals who did not have a stroke by age 75, twenty
percent had high blood pressure at age 70” becomes
Pr{H | S c} = 0.20
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A Probability Tree for the Example
Collecting everything we know:
Pr{S} = 0.10
Pr{H | S} = 0.40
Pr{H | S c} = 0.20
We can expand our tree to
0.1
0.9
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0.4
H
0.04
0.6
Hc
0.06
0.2
H
0.18
0.8
Hc
0.72
S
Sc
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Example (cont.)
0.1
0.9
0.4
H
0.04
0.6
Hc
0.06
0.2
H
0.18
0.8
Hc
0.72
S
Sc
Note that the probabilities show on the branches are conditional.
To obtain the unconditional probabilities of the final nodes
multiply along the branches.
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Example (cont.)
The questions of interest are:
1. What is the probability that a 70 year-old patient has high
blood pressure?
We can calculate this as:
Pr{H} = Pr{H&S} + Pr{H&S c} = 0.04 + 0.18 = 0.22
This is the Rule of Total Probability.
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Example (cont.)
2. What is the probability that a 70 year-old patient with high
blood pressure will have a stroke within five years?
Notice that in this question, the order of conditioning is reversed.
We want Pr{S|H} but we have only been given information on
Pr{H|S} and Pr{H|S c}.
However, there is a formula we can use:
Pr{H and S}
= 0.04/.22 = 0.182
Pr{S|H} =
Pr{H}
This is the Conditional Probability Rule.
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