Key Concepts of the Probability Unit

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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!

8. 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

9. Example of Conditional Probability

10. 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:

11. 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