Download STAB22 Statistics I Lecture 16 1

STAB22 Statistics I
Lecture 16
1
Probability Tree

Describe events & probabilities at different stages
Conditional
Probabilities
Simple
Probabilities
Joint
Probabilities
B
P(BC|A)
BC
P(A ∩ BC) = P(A) × P(BC|A)
B
P(AC ∩ B) = P(AC) × P(B|AC)
BC
P(AC ∩ BC) = P(AC) × P(BC|AC)
A
P(A)
P(B|AC)
P(AC)
P(A ∩ B) = P(A) × P(B|A)
P(B|A)
AC
P(BC|AC)
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Example (Monty Hall Problem)




Contestant shown 3 doors:
 1 contains a car
 other 2 contain goats
Contestant picks a door
Presenter must open a door
containing a goat,
Then offer contestant to
 Switch doors, or
 Stay with his original pick
( From TV show “Let’s Make a Deal” )
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Example (Monty Hall Problem)
4
Example

Doctor prescribes screening test for Hepatitis B
(HB), since it affects 2% of population


What is prob. of having HB? P(HB) =
However, the test is not perfectly accurate!


If you have HB, test is positive (+) with P(+|HB) = 96%
If you don’t (HBC), test is negative (−) with P(−|HBC) = 98%
Actual


Test’s Accuracy table:
Prob Correct
HB
96%
HBC
98%
Build a probability tree for the test
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Accuracy table:
Example
Simple
Probs
Actual
Condition
HB
HBC
Conditional
Probs
Actual
Prob Correct
HB
96%
HBC
98%
Test
Result
Joint Probs
+
−
+
−


Find prob. of testing positive: P(+) =
Find prob. of having HB given that you tested positive:
P(HB|+) =
6
Random Variables

Consider experiment of rolling 2 dice

How can you define event A?
1.
Describe the event

2.
3.
Event A
A = “Sum of two rolls is 3”
List outcomes in event
 A = {(1,2),(1,2)}
Use a Random Variable

Most common way
to describe events!
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
Sample space (S)
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Random Variables

Random variable X assigns a single number
to every outcome of an experiment


Eg. Consider flipping 2 fair coins
Random variable X = # of heads in 2 flips
Sample space (S)
 H , H   H ,T 
 T , H  T , T 
Values of random variable X
X  H ,H   2
X  H ,T   1
X T , H   1
X T ,T   0
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Random Variables


In practice, Random Variables used to
describe events and their probability
Example: Rolling 2 fair dice

Random Variable X = sum of 2 rolls

List sample points of event
(X=5)

Find probability P(X=5)
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
Sample space
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Types of Random Variables

Discrete
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Can only take specific value, e.g. 0, 1, 2, ...
Typically the result of counting something


E.g. the number of children in a family
Continuous


Can take any value within an interval, e.g. all
values in [0,1]
Typically the result of some type of measurement

E.g. the temperature tomorrow
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Probability Model

A table, graph, or formula that describes



The values of a random variable
The probability associated with each value
Probability of X taking some specific value x is
denoted by P(X=x) or just P(x)


E.g. X = sum of 2 dice rolls → P(X=5) = P(5)= 4/36
For any probability model:
0  P ( X  x)  1 &
 P ( X  x)  1
all x
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Example

Flip 2 fair coins, X = # of heads

Fill in the probability model
Value x
P(X=x)
X  H ,H   2
X  H ,T   1
X T , H   1
X T ,T   0
Sample space
Total
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Example

Flip 3 fair coins, X = # of heads
Probability model →

Find prob. of getting 2 or more heads

Find prob. of getting at least 1 head
Value x
P(X=x)
0
1/8
1
3/8
2
3/8
3
1/8
Total
1
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Expected Values

Probability model of (discrete) Random
Variable can be described numerically with:

Mean or Expected Value:   E  X    xP  x 
all x

2
2
Variance:   Var ( X )    x    P  x 
all x

Standard Deviation:   SD( )  Var ( X )
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Example

Flip 2 fair coins, X = # of heads

Find mean, variance, and standard deviation of X
x
P(x)
0
1/4
1
2/4
2
1/4
Total
1
x × P(x)
(x − µ)²
(x − µ)² × P(x)
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