Special Lecture: Conditional Probability

1. 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.) Don’t forget to sign in for credit!

2. The Plan… Today, I plan to cover material related to these ALEKS topics. Specifically, we’ll… • Review all the formulas we’ll 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.

3. Formulas:

4. Formulas: p(A|B) = p(B|A)*p(A) p(B) Bayes’ Theorem: This is simply derived from what we already know about conditional probability. Or if we don’t have p(B) we can use the more complicated variation of Bayes’: p(A|B) = p(B|A)*p(A) p(B|A)*p(A) +p(B|A’)*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(A|Bn)p(Bn)

5. Formulas: All together now

6. 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 let’s do that…

7. Shapes Demo Imagine that we have the following population of shapes: = 8/24 = 1/3 = 8/24 = 1/3 = 8/24 = 1/3 • 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

8. Now that we’ve figured out the probability of these events, What else can we do? = 8/24 = 1/3 = 8/24 = 1/3 = 8/24 = 1/3 • 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

9. Now that we’ve 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’s the probability of getting a blue triangle? • P( ) • P( ) • P() • P() = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 1/4 p( ) = p(B ) = 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

10. Now that we’ve 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( ) = p(B ) = 1/12 • P( ) • P( ) • P() • P() = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 1/4 p( or B or ) = p(B ) = p(B )+p( )- p(B ) = 8/24 +6/24 - 1/12 =12/24 =1/2 • P(BIG) • P(small) = 12/24 = 1/2 = 12/24 = 1/2

11. Now that we’ve 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( ) = p(B ) = 1/12 • P( ) • P( ) • P() • P() = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 1/4 = 6/24 = 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) = 12/24 = 1/2 = 12/24 = 1/2 = 2/24 / 8/24 = 2/8 = 1/4

12. So, the calculations work out… But do they make sense??

13. 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.

14. So, Let’s Try an ALEKS problem.

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26. Any other questions or concerns?