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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. I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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Presentation Transcript 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
• 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
• 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
• 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
• 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
• 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
• 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 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 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
• 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