1. 1 Chapter 4, Part 1 Basic ideas of Probability Relative Frequency, Classical Probability Compound Events, The Addition Rule Disjoint Events

2. 2 Idea of Probability Probability is the science of “random” phenomena or “chance” behavior. Many phenomena are unpredictable when observed only once, but follow a general pattern if observed many times. Examples of “random” phenomena: –Coin flips, rolling dice, drawing cards –Drawing names/numbers out of a hat –** Choosing a sample from population **

3. 3 Modeling Random Behavior When we observe “random” behavior, we do not try to predict the results of a single observation. Instead, we… –Consider every possible result that could happen on a given observation. These are called outcomes for the random phenomenon. –Measure the “chance” that a given outcome will occur on a particular observation. This is called a probability.

4. 4 Basic Terminology An event is any set of outcomes or results of a given random phenomenon. An outcome (or simple event) is a single result that cannot be broken down into simpler components. The Sample Space for a random phenomenon is the set of all possible outcomes.

5. 5 Basic Terminology, Examples I randomly select one student from my class list. Some (non-simple) events: –The student is in Row 1 –The student is absent –The student is female –The student’s Exam 1 score was above 90%. Each individual student on the class list is an outcome (simple event). The Sample Space is the set of all students on my class list.

6. Probability When we try to model/describe a random phenomenon, each event is assigned a number, called the probability of the event. Probability measures how likely it is that an event will occur. Events with higher probability are more likely to happen (or tend to happen more frequently) P(A) denotes the probability of event A. We always require that 0 ≤ P(A) ≤ 1.

7. Possible Values for Probabilities: 0 ≤ P(A) ≤ 1

8. Three Views of Probability Relative Frequency (Empirical Probability): –Actual Data is used to estimate the probability of various events. Classical Approach (Theoretical Probability): –Assign probabilities in a way that satisfies a set of formal mathematical rules. Subjective Probability (“Expert Opinion”): –Use prior knowledge from a similar situation in order to estimate probabilities.

9. 9 Relative Frequency Estimate probability from actual data. Take many observations of a random phenomenon, and count how many times a particular event occurs. The relative frequency of the event is: (# of occurrences) / (# of observations). In other words, for what proportion of observations did the event occur? It may be helpful for you to think of this as a percentage.

10. Relative Frequency: Examples Earlier in the class, you drew a slip of paper from the hat. This gives us about 30 observations of a random phenomenon. Using our actual data, we now compute: Relative frequency of “Blue” = Relative frequency of “Pink” =

11. Classical Probability Our random phenomenon has Sample Space with finitely many outcomes, say n of them. Assume that each outcome is equally likely to occur on any given observation. The (classical) probability of the event is: (# of outcomes in the event)/(total # of outcomes) NOTE: This is actually just a special case of a more general approach (theoretical probability).

12. Classical Probability, Examples I randomly select one student from class (among those currently present). Assume that each student (outcome) is equally likely. Compute the (classical) probability of: –The student is in Row 1. –The student is in the back row. –The student is texting on his/her cell phone.

13. Why Classical Probability? Suppose our random phenomenon is “Choose a sample of N individuals from a large population.” Each outcome is a group of N individuals. Note that N is NOT the total number of outcomes (that number is MUCH BIGGER) If we assume that “all outcomes are equally likely,” then we are talking about…?

14. 14 The Law of Large Numbers Relative Frequency: Estimate probability using actual observational data. Classical Probability: Compute using knowledge of the Sample Space. Question: What if our knowledge of the Sample Space is incomplete? –Example: We know the hat has only pink and blue slips, but we don’t know how many of each kind.

15. 15 The Law of Large Numbers Under a certain condition (independent observations, discussed next time): As we increase the number of observations, the Relative Frequency of an event tends to be closer to the (theoretical) Probability of that event. Relative Frequency estimates Probability. With more observations, you are more likely to get a better estimate.

16. Compound Events Let A, B be two events. For example, if I choose a student from those in class: –A = “The student I choose is male.” –B = “The student I choose is in the back row.” Each event is actually a set of outcomes, but you can think of each event as some kind of condition/requirement. The event “A or B” is the set of outcomes that meet at least one (but possibly both) of the given requirements.

17. Compound Events The event “A and B” is the set of outcomes that meet both of the given requirements. In the previous example: –“A or B”: The chosen student is male, in the back row, or both (at least one condition is met). –“A and B”: The chosen student is male AND in the back row (both conditions are met).

18. 18 The Addition Rule If A and B are any events, then we have: P(A or B) = P(A) + P(B) – P(A and B) Here’s a version that’s useful when “all outcomes are equally likely”: Let #(A) be the number of outcomes in event A. Then #(A or B) = #(A) + #(B) - #(A and B)

19. Disjoint Events If events A and B have no outcomes in common (they cannot occur at the same time), then we say that they are disjoint. –Example: “Student is in Row 1” and “Student is in Row 3” In this case P(A and B) = 0. So the Addition Rule becomes: P(A or B) = P(A) + P(B)

20. Complementary Events Given an event A, the complement of A is the set of outcomes that are not in A. –Notation: is the complement of A. Example: Choose a student from class: –A = “Student is in Row 1” –B = “Student is Female” Note that an event and its complement will always be disjoint.

21. Formulas for Complementary Events