Copy of Chapter 5 Notes Outline - Probability in Our Daily Lives PDF

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Chapter 5 Probability in Our Daily Lives Please use this notes page as a guide and fill in any blanks, definitions or extra notes as you read the textbook and view the lecture presentation. Note: As you view the lecture presentation, it is very helpful to have the notes tab showing to assist in filling in the blanks.

5.1 How Probability Quantifies Randomness Definitions (slide 2): Randomness - We know the possible outcomes of a response variable (for example, died of a heart attack, yes or no, or rolling a die can result in 1, 2, 3, 4, 5 or 6) but what is random is the outcome that actually occurs for any given observation. Trial - each outcome of an event or experiment, for example, one roll of a die or spin of a spinner or toss of a coin. Cumulative proportion - A running record of the p  roportion of times that an outcome has occurred (slide 4) With random phenomena, the proportion of times that something happens is highly random and variable in the short run but very predictable in the long run. More Definitions (slide 5): Independent trials - Trials are independent of each other if w  hat happens on previous trials does not affect the trial about to occur. Probabilities take on values between 0  and 1 when written as proportions and between 0 and 100 when written as p  ercentages. Subjective Probabilities - probabilities that are not based on a long run of trials. Subjective probabilities are a personal probability, your degree of belief that an outcome will occur based on all of the available information.

5.2 Finding Probabilities Definitions (slide 6): ● Sample space - T  he set of all possible outcomes. For example, when rolling a die the sample space is 1, 2, 3, 4, 5, and 6 and when tossing a coin, the sample space is heads or tails. ● Event - a subset of sample space. For example, when rolling a die, an event could be rolling a 4 or an event could be rolling an odd number. ●

Probabilities must have the following two properties: ○ The probability of each individual outcome is between 0 and 1. ○ The s  um total of all of the individual probabilities for a sample space equals 1.

Example (Slide 7): Suppose a student takes a 3-question pop quiz. Each response might be graded as correct or incorrect. 1. What is the sample space for this experiment? (Sketch the tree diagram sample space here) 2. What is the probability of each outcome? 8 total outcomes, the probability of anyone of those outcomes is ⅛=0.125 3. Let A = “exactly 2 correct.” What are the outcomes in the event A? The outcomes in Event A, “exactly 2 correct” would be CCI, CIC, ICC 4. Let B = “passes (with 2 or more correct).” What are the outcomes in the event B? The outcome in Event B passes “(2 or more correct)” would be CCC, CCI, CIC, ICC

Basic Rule of Probability of an Event (A) (Slide 8)

Basic Rules for Finding Probabilities about a Pair of Events Consider two events A & B in a sample space S. Draw your own Venn Diagram sketch for each. 1. Disjoint Events (Slide 10): If two events A and B have no outcomes in common they are called disjoint events. Then the probability that one or the other occurs is the sum of their individual probabilities. That is P(A or B)= P(A)+P(B). 2. Union (A or B) (Slide 11) -Outcomes that are in A or B or both. In a Venn diagram, this would be everything inside of both events A and B. In general, P(A or B) = P(A⋃B) = P(A) + P(B) - P(A and B). Note on Slide 12 there is a typo and the sum for part (d) should be 450 + 400 - 150 = 700, not 800. 3. Intersection (A & B) - (Slide 14) O  utcomes that are in both A and B. In a Venn diagram, this is where the two sets overlap. For disjoint events, P(A and B) = P( A∩B) = 0 Disjoint events do not overlap so there are no common outcomes with disjoint events.

Addition Rule: Finding the Probability that Event A or Event B Occurs (Slide 13) Remember, this is used when we are f inding OR probabilities. Addition Rule P(A or B) = P(A) + P(B) - P(A and B)

Multiplication Rule: Finding the Probability that Events A and B Both Occur (Slide 16). This is the intersection of A and B, both events have to happen. This rule would be used for probabilities that use the word AND.

Independence: If two events are independent, knowledge about one event t ells us nothing about the other event. What happens on one trial is not influenced by what happens on any other trial. Some examples of independent events are rolling a die, flipping a coin, winning a checkers game, making a basket in basketball

5.3 Conditional Probability: The Probability of A Given B Conditional probability (Slide 2) deals with finding the probability of an event when you know that the outcome was in s  ome particular part of the sample space. You have “narrowed down” the sample space and are no longer considering the entire sample space

5.33 Revisiting seat belts and auto accidents (Slide 6) The following table classifies auto accidents by survival status (S = survived, D=died) and seat belt status of the individual involved in the accident.

The three most important things to remember with conditional probability: The three most important things to remember with conditional probability: 1. You are not looking at the entire sample with a conditional probability problem, you are just looking at a  smaller part that meets a certain condition. 2. If you see the word “given” in the problem, this will be a conditional probability problem. You are n  arrowing down the entire sample to a smaller subset of the sample that has the “given” condition. 3. A conditional probability problem will not always contain the word “given”, but a condition can be implied in the problem. Conditional Probability Formula P(A | B)= P(A and B) / P(B)

Sampling With Replacement and Sampling Without Replacement (Slide 10) In many sampling processes, once subjects or objects are selected from a population, they are not eligible to be selected again. This is called s  ampling without replacement. When sampling without replacement is used, probabilities of upcoming o  utcomes depend on the previous outcomes. Sampling Without Replacement Example (Slide 11):

Sampling With Replacement Example (Slide 12): What is the probability of a monkey typing macbeth on a keyboard with 50 keys? Answer: Notice the difference between with replacement and w  ithout replacement. In both cases, you are doing the same thing multiple times and you are m  ultiplying the probabilities. When you have with replacement, like this monkey example, the individual probabilities stay the same. When you have without replacement, like with the card example on the previous slide, the probabilities adjusted a bit with each successive outcome, (10/47)(9/46), they were going down in number. Example problem: 5.39 Happiness in marriage (Slide 13 and 14) Are people happy in their marriages? The table shows results from the 2008 General Social Survey for married adults classified by gender and level of happiness. a. There are 398 very happy people out of 969 surveyed so the probability P(VH)=0.411

b.P(VH/M)=183/469=0.390 C. Events are not independent

c....