Key Concepts of the Probability Unit Simulation Probability

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!

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 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 (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