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 (AP optional) 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 One of the reasons we spend so much time thinking about or drawing out a sample space is so that probability is more about accurate counting (and some vocab: and, or, given) than seemingly mysterious formulas ALWAYS SHOW THE FRACTION TO SHOW YOUR COUNTS BEFORE CALCULATION PROBABILITY! 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) Use with a series of decisions or events Use to build probability distributions (ch 16)

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 not independent. 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 or mutually exclusive.. 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 chapter 4/5 Can be calculated from other probabilities using this formula: Example: P(Draw a red card 2 nd, given a red card was drawn 1 st ) is equal to P(red card 1 st x red card 2 nd )/P(red card 1 st ), which equals

Example of Conditional Probability

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 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