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Key Concepts of the Probability Unit. Simulation Probability rules Counting and tree diagrams Intersection (“and”): the multiplication rule, and independent events Union (“or”): the addition rule, and disjoint events Venn diagrams Conditional probability and Bayes Rule. Simulation.

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key concepts of the probability unit
Key Concepts of the Probability Unit
  • Simulation
  • Probability rules
    • Counting and tree diagrams
    • Intersection (“and”): the multiplication rule, and independent events
    • Union (“or”): the addition rule, and disjoint events
    • Venn diagrams
    • Conditional probability and Bayes Rule
simulation
Simulation
  • Can often be used to estimate probabilities, especially when there is a complex series of events
  • Is a valid technique for verifying the results of a probability model
  • Is accepted on the AP Exam
  • Can be done using a calculator, computer, or random number table
counting
Counting
  • It is necessary to determine how many outcomes are in a sample space before we can determine the probability of an event
  • Usually requires determining how many ways each part of an event can happen, then finding the product of these
  • Counting problems usually involve combinations and permutations, concepts that are (surprisingly) not covered in this book
tree diagrams
Tree Diagrams
  • Very useful for illustrating and determining how many ways outcomes can occur (how many items are in a sample space)
  • Can also be used to calculate the associated probability of each outcome
intersection
Intersection
  • The intersection of P(A) and P(B), means the probability of both A and B occurring, and is denoted by
    • If the outcome of event A has no impact upon the outcome of event B, they are said to be independent. Calculating then is very easy, it is just P(A) x P(B).
      • Example: probability of rolling a “6” on a die, then drawing a “red” card.
    • If the outcome of event A has an impact upon the outcome of event B, they are said to be dependent. Calculating then is more involved: it is P(A) x P(B/A), read as Probability of B given A.
      • Example: probability of drawing a red card, then drawing another red card/given that the first card was red
union
Union
  • The union of P(A) and P(B), means the probability of A or B occurring, and is denoted by
    • If the outcome of event A has no possibility of occurring at the same time as event B, they are said to be disjoint. Calculating then is very easy, it is just P(A) + P(B).
      • Example: probability of rolling a “6” on a die or rolling a “3”.
    • If the outcome of event A can occur at the same time as event B, they are said to be not disjoint. Calculating then is more involved, it is P(A) + P(B) –
      • Example: probability of rolling a “greater than 3” on a die or rolling an “even number”: P(greater than 3) + P(even) – P(4 or 6)
venn diagrams
Venn Diagrams
  • Very useful for Intersection and Union problems
  • Visual displays of Intersection, Union, and Complementary probabilities
  • Re
  • Remember that P(D) is equal to the sum of the light green and blue regions!
  • P(D) is equal to the sum of the light green and blue regions!
conditional probability
Conditional Probability
  • Conditional probabilities are a logical next step from the Conditional Distributions we studied in 4.2
  • Can be calculated from unconditional probabilities using this formula:
  • Example: P(Draw a red card 2nd, given a red card was drawn 1st ) is equal to P(red card 1st x red card 2nd)/P(red card 1st), which equals
bayes rule
Bayes Rule
  • Bayes rule allows us to calculate P(B/A) if we know P(A/B)
  • Often it is easier to derive P(B/A) without using Bayes Rule by using a Tree Diagram (see textbook Ex. 6.31)
  • Bayes Rule:
example of bayes rule
Example of Bayes Rule
  • From our previous example, we saw that P(“A”/liberal arts) was 34%. Can we use the information we have to find P(liberal arts/“A”)? Recall that…
  • So, P(lib arts/A) = P(A/lib arts)P(lib arts)
                  • P(A)

(.34)(.63)/.34 = .6314