Special Lecture: Conditional Probability

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Special Lecture Conditional Probability
Example of Conditional Probability in the real
world This chart is from a report from the
CA Dept of Forestry and Fire Prevention. It
shows the probability of a structure being lost
in a forest fire given its location in El Dorado
county. (calculated using fuel available, land
slope, trees, neighborhood etc.)

• Dont forget to sign in for credit!

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The Plan

• Today, I plan to cover material related to these
  ALEKS topics.

• Specifically, well
• Review all the formulas well need.
• Go over one conceptual example in depth.
• Work through a number of the ALEKS problems that
  have been giving you trouble.
• Address any specific questions/problems.

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Formulas
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Formulas

• Bayes Theorem
• This is simply derived from what we already know
  about conditional probability.

p(AB) p(BA)p(A) p(B)
Or if we dont have p(B) we can use the more
complicated variation of Bayes
p(AB) p(BA)p(A)
p(BA)p(A) p(BA)p(A)
The reason those two formulas are the same has to
do with the Law of Total Probabilities For
any finite (or countably infinite) random
variable,
p(A) ? p(A?Bn) or, p (A) ?
p(ABn)p(Bn)
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Formulas All together now
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Shapes Demo
Imagine that we have the following population
of shapes

• Notice that there are several dimensions that we
  could use to sort or group these shapes
• Shape
• Color
• Size
• We could also calculate the frequency with which
  each of these groups appears and determine the
  probability of randomly selecting a shape with a
  particular dimension from the larger set of
  shapes.
• So lets do that

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Shapes Demo
Imagine that we have the following population
of shapes
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• P(R)
• P(Y)
• P(B)

• P( )
• P( )
• P( )
• P( )

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1/4

• P(BIG)
• P(small)

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Now that weve figured out the probability of
these events, What else can we do?
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• P(R)
• P(Y)
• P(B)

• P( )
• P( )
• P( )
• P( )

6/24 1/4 6/24 1/4 6/24 1/4 6/24
1/4

• P(BIG)
• P(small)

12/24 1/2 12/24 1/2
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Now that weve figured out the probability of
these events, What else can we do? Lots of
stuff!
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• P(R)
• P(Y)
• P(B)

Whats the probability of getting a blue triangle?

• P( )
• P( )
• P( )
• P( )

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1/4
p(B? )
p( )
p(B)p( )
8/24 6/24 48/576
2/24 1/12

• P(BIG)
• P(small)

12/24 1/2 12/24 1/2
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Now that weve figured out the probability of
these events, What else can we do? Lots of
stuff!
8/24 1/3 8/24 1/3 8/24 1/3

• P(R)
• P(Y)
• P(B)

What else?
p(B? ) 1/12
p( )

• P( )
• P( )
• P( )
• P( )

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1/4
p( or B or )
p(B? )
p(B )p( )- p(B? )
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1/2

• P(BIG)
• P(small)

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Now that weve figured out the probability of
these events, What else can we do? Lots of
stuff!
8/24 1/3 8/24 1/3 8/24 1/3

• P(R)
• P(Y)
• P(B)

What else?
p(B? ) 1/12
p( )

• P( )
• P( )
• P( )
• P( )

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1/4
p( or B or )
p(B? )1/2
p( given that we have B)
p( B)
p(B? ) /p(B)

• P(BIG)
• P(small)

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2/24 / 8/24 2/8
1/4
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So, the calculations work out
But do they make sense??
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How to approach ALEKS problems

• Write down everything you know.
• Write down (and probably draw out) what you need
  to figure out.
• Figure out a plan.
• Go.

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So, Lets Try an ALEKS problem.
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Any other questions or concerns?