Chapter 4 Probability Notes

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Chapter 4 Probability Notes

  1. 1. CHAPTER 4 - PROBABILITY
  2. 2. INTRODUCTORY VOCABULARY <ul><li>Random (trials) – individual outcomes of a trial are uncertain, but when a large number of trials are performed a regular distribution appears </li></ul><ul><li>Probability – Proportion of times an outcome would occur in a large number of trials </li></ul><ul><li>Experimental Probability – What did happen in an experiment. The proportion of times an event occurred in an experiment </li></ul>
  3. 3. <ul><li>Theoretical Probability – What should happen in an experiment. Usually found by looking at experimental probabilities. </li></ul><ul><li>Probability Models – </li></ul><ul><ul><li>List of all possible outcomes </li></ul></ul><ul><ul><li>The probability of each outcome is then listed. </li></ul></ul><ul><li>Sample Space – the set of all possible outcomes of an event. S = { } </li></ul><ul><li>Examples: </li></ul><ul><ul><li>Rolling a die once; S = {1,2,3,4,5,6} </li></ul></ul><ul><ul><li>Flipping a coin twice; S = {HH,HT,TH,TT} </li></ul></ul>
  4. 4. PROBABILITY NOTATION <ul><li>A,B,C, etc. – events or outcomes </li></ul><ul><li>P(A) = the probability of outcome A occuring </li></ul><ul><li>S = sample space </li></ul><ul><li>When we represent events, we draw them with Venn Diagrams </li></ul><ul><li>Venn Diagrams use shapes to represent events and a box around the shapes that represents the sample space or all possible outcomes </li></ul>
  5. 5. GENERAL SET THEORY <ul><li>Union: “or” statements </li></ul><ul><li>Meaning: joining, addition </li></ul><ul><li>Symbol: </li></ul><ul><li>Example 1: </li></ul><ul><li>Example 2: Set A = {2,4,6,8,10,12} </li></ul><ul><li> Set B = {1,2,3,4,5,6,7} </li></ul><ul><li>A B = </li></ul>A B
  6. 6. <ul><li>Intersection: “and” </li></ul><ul><li>Meaning: overlap, things in common </li></ul><ul><li>Symbol: </li></ul><ul><li>Example 1: </li></ul><ul><li>Example 2: Set A = {2,4,6,8,10,12} </li></ul><ul><li> Set B = {1,2,3,4,5,6,7} </li></ul><ul><li>A B = </li></ul>A B
  7. 7. <ul><li>Complement: of event A </li></ul><ul><li>Meaning: not A. None of the outcomes of event A occur. Everything but A </li></ul><ul><li>Symbol: A C </li></ul><ul><li>Example 1: Shade A C Shade A C B </li></ul><ul><li>Example 2: Set A = {2,4,6,8,10,12} </li></ul><ul><li>S = {whole numbers 1 to 15} </li></ul><ul><li>A C = { </li></ul>A B A B
  8. 8. TRY THE SET THEORY WORKSHEET
  9. 9. PROBABILITY RULES! <ul><li>First Three Probability Rules </li></ul><ul><li>All probabilities lie between 0 and 1 </li></ul><ul><li>Probability of all possible outcomes must be equal to 1 </li></ul><ul><li>Probability of the compliment of A is the same as 1 minus the probability of A </li></ul><ul><li>Example 1: </li></ul>
  10. 10. <ul><li>Example 2: </li></ul><ul><li>Example 3: </li></ul>Type A+ A- B+ B- AB+ AB- O+ O- Prob. 0.16 0.14 0.19 0.17   ? 0.07 0.1 0.11
  11. 11. <ul><li>Unions </li></ul><ul><li>OR => Addition </li></ul><ul><li>General Rule: </li></ul><ul><li>Why do we subtract the intersection? </li></ul><ul><li>We don’t want to count the outcomes in A and B twice, the overlap of A and B. </li></ul>A B
  12. 12. <ul><li>Special Case: </li></ul><ul><li>What if A and B don’t overlap? </li></ul><ul><li>So </li></ul><ul><li>This is called Disjoint or Mutually Exclusive </li></ul><ul><li>No common outcomes </li></ul>
  13. 13. <ul><li>Conditional Probability </li></ul><ul><li>Probability of B happening given that A has already happened. </li></ul><ul><li>Formula: </li></ul><ul><li>Example: </li></ul><ul><li>P(A) = 5/10 </li></ul><ul><li>P(B) = 3/10 </li></ul><ul><li>P(B|A) = 3/9 since the first one was not replaced </li></ul><ul><li>P(B|A)=P(A|B)?? </li></ul>
  14. 14. <ul><li>Intersections </li></ul><ul><li>General Rule: </li></ul><ul><li>Also called the multiplication rule </li></ul><ul><li>Special Case </li></ul><ul><li>P(Red) = 3/10 P(Red|Blue) = 3/10 </li></ul><ul><li>If P(B|A) = P(B) the two events are independent </li></ul>

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