Notes Probability AP Statistics Probability: A branch of mathematics that describes the pattern of chance outcomes. Probability outcomes are the basis for inference. Randomness: (not haphazardous) A kind of order that emerges in the long run when repeated events occur. We call a phenomenon random if individual outcomes are uncertain, but there is none-the-less a regular distribution of outcomes in a large number of repetitions. Experiment: Any sort of activity whose outcome cannot be predicted with certainty. (Flip a coin, roll a die, etc.) Outcome: One of the possible things that can occur as the result of an experiment. Sample Space (S): The set of all possible outcomes. Example: Flipping Coins A Tree Diagram can help define our sample space (S). Ways to Calculate Outcomes Fundamental Counting Principle: If there are m different choices for decision 1 and n different choices for decision 2, then the first and second decisions together can be taken m x n ways. Combinations: Groupings without ordering. (Order does not matter) n objects taken r at a time. Permutations: Grouping by order. (Order matters) Examples: On a menu, there are 5 appetizers, 10 entrees, 6 desserts, and 4 beverages. How many possible dinners are there? ABCDE How many groups are possible if you have 5 letters taken 2 at a time? ABCDE How many two letter words can be made from these 5 letters taken two at a time.
Probability of an Outcome: A number that represents the likelihood of the occurrence of an outcome. Theoretical Probability: probability values arrived at through calculations. Empirical Probability: probability values arrived at through observation or simulation. Law of large numbers: the more trials you do the closer your Empirical probability gets to the Theoretical probability. Basic Rules for Probability The probability P(A) of any event A satisfies 0 < P(A) < 1 P(A) = 0 if and only if A is certain not to occur (impossible event). P(A) = 1 if and only if A is certain to occur. If S is the sample space in a probability model, then P(S) = 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula number of elements in A PA number of elements in S The complement of any event A is the event that A does not occur, written as A c. The complement rule states that P(A c ) = 1 P(A). The At Least One Rule: P(At least one) = 1 P(none) The Addition Rule for Disjoint Events If A and B are disjoint (can t happen at the same time), then: P(A or B) = P(A) + P(B) P(A U B) = P(A) + P(B) EX. Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following is the distribution of similar preferences: Number of Similar Preferences All four Three Two One None Probability.09.35.33.19 a. What is the sample space? b. What is the probability that a married couple had no preferences in common? c. What is the probability that a married couple had at least one preference in common? d. What is the probability that a married couple had no more than two preferences in common? e. What is the probability that a married couple had at least two preferences in common? Name two ways to make this calculation. f. What is the probability that a married couple had either all four or no preferences in common?
Two events are independent when knowing that one occurred does not change the probability that the other occurred. Multiplication Rule for Independent Events If two events are independent, then the probability of both A and B occurring is: P(A and B) = P(A) P(B) P(A B) = P(A) P(B) EX. Compute the probability of the following situations based on a standard deck of cards. a. Drawing two aces with replacement. b. Drawing 3 face cards with replacement. c. Draw three odd numbered red cards with replacement. General Rule for the Union of Two Events For any two events A and B, P(A or B) = P(A) + P(B) P(A and B). P(A U B) = P(A) + P(B) P(A B) Suppose that 60% of all customers of a large insurance agency have automobile policies with the agency, 40% have homeowner s policies, and 25% have both types of policies. If a customer is randomly selected, what is the probability that he or she has an automobile policy or a homeowner s policy? What percent will have neither? Medical records indicate that the probability that a man over 50 years of age in a mining community has black lung disease is 0.35. The probability that the individual has high blood pressure is 0.21. We also know that the chance of them having black lung disease and high blood pressure is 0.18. What is the probability that a man over 50 years of age in a mining community has black lung disease or high blood pressure?
Conditional Probability: The probability that one event happens given that another event is already known to have happened. P(B A) is the conditional probability that B occurs given the information that A occurs. Read as the probability of B given A. General Multiplication Rule: The joint probability that both of two events A and B happen together can be found by P(A and B) = P(A) P(B A) P(A B) = P(A) P(B A) EX. Compute the probability of the following situations based on a standard deck of cards. a. Drawing two even numbered cards without replacement. b. Drawing 5 red cards without replacement. A GFI (ground fault interrupt) switch will turn off power to a system in the event of an electrical malfunction. A spa manufacturer currently has 25 spas in stock, each equipped with a single GFI switch. The switches are supplied by two different companies, and some of them are defective. Company Company Nondefective Defective 1 10 5 2 8 2 a. What is the probability of a GFI switch from a selected spa is from company 1? b. What is the probability of a GFI switch from a selected spa is defective? c. What is the probability of a GFI switch from a selected spa is defective and from company 1? d. What is the probability of a GFI switch from a selected spa is from company 1 given that it is defective? e. What is the probability of a GFI switch is defective knowing the switch is from Company 2?
Independent Events Recall the Multiplication Rule for Independent Events: If two events are independent, then the probability of both A and B occurring is: P(A and B) = P(A) P(B) The General Multiplication Rule states that P(A and B) = P(A) P(B A) So that means if the events are independent, then P(B) = P(B A) and then if P(B) = P(B A) then the events must be independent EX. Pick a card, replace/reshuffle, pick another. What is the probability that the first card is red and the second is a king? Jack and Jill have finished conducting taste tests with 100 adults from their neighborhood. They found that 60 of them correctly identified the tap water. The data is displayed below. Identified Tap Water Yes No Total Male 21 14 35 Female 39 26 65 Total 60 40 100 Is the event that a participant is male and the event that he correctly identified tap water independent?
Calculating Conditional Probabilities Recall the General Multiplication Rule Divide both sides by P(A) P(A and B) = P(A) P(B A) We can use this new formula to find the conditional probability P(B A). One exciting aspect of increased communication using the Internet is that diverse individuals from widely scattered places all over the world can form an electronic chat room. A side effect of such conversation is negative criticism of others contributions to the conversation. Investigators are interested in the effect that personal criticism has on an individual. Would being criticized make one more likely to criticize others? Have Criticized Others Have Not Criticized Others Have Been Personally Criticized Have Not Been Personally Criticized 19 8 23 143 We will assume that the table from the article is indicative of the larger group of chat room users. Suppose that a chat room user is randomly selected. Let C = event that the individual has criticized others and let O = event that the individual has been personally criticized by others. Write the following events out in words and then solve the probability. P( C) P( O) P( CO) P( C O) P( OC)
Conditional Probability with Tree Diagrams A tree diagram can be used to model chance behavior that involves a sequence of outcomes. The events may be independent or dependent. To find the probabilities you multiply down the branches. Ex. Dr. Carey has two bottles of sample pills on his desk for the treatment of arthritic pain. He often grabs a bottle without looking and takes the medicine. Since the first bottle is closer to him, the chances of grabbing it are 0.60. He knows the medicine from this bottle relieves the pain 70% of the time while the medicine in the second bottle relieves the pain 90% of the time. What is the probability that Dr. Carey grabbed the first bottle given his pain was not relieved? Ex. Due to the rising costs of auto insurance, the probability that a randomly selected driving in one particular city drives an uninsured motor vehicle is 0.13. Knowing the driver is uninsured leads to the driver s likelihood of being under 30 to be 0.72 while the percent drops to.58 for those drivers who are insured. What is the probability that a driver selected at random is uninsured given the driver are under 30?