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AP Statistics. Notes Chapter 14 and 15. Notes Chapter 14. Probability is the branch of mathematics that describes the pattern of chance outcomes. The probability of an outcome is the proportion of times we expect the outcome would occur in a very long series of repetitions.
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AP Statistics Notes Chapter 14 and 15
Notes Chapter 14 • Probability is the branch of mathematics that describes the pattern of chance outcomes. • The probabilityof an outcome is the proportion of times we expect the outcome would occur in a very long series of repetitions.
For any random variable, each attempt, or trial, generates an outcome. • The sample space, S, of a random variable is the set of all possible outcomes for the variable. • Sampling can be done with or without replacement based on context. When we draw from a set, do we replace after each draw or not. • An event is an outcome or a set of outcomes of a random variable.
Disjoint events (also called mutually exclusive) have no outcomes in common. They can not occur at the same time. • Events are independent (informally) when the outcome of one event does not influence the outcome of any other event. • Be certain not to confuse disjoint with independent. Disjoint events cannot occur together, therefore they are never independent.
The Law of Large Numbers says that as the number of trials increases, the experimental probability approaches the theoretical probability. • A probability distribution for a certain variable is a table made up of the sample space and the individual probabilities of each event in the sample space. • The following three rules apply to probability distributions.
Rule 1: The probability of any event is a number between 0 and 1. 0 P(x) 1. • Rule 2: The sum of all possible outcomes must equal 1. P(S) = 1. • Rule 3: The probability that an event does not occur is 1 minus the probability that the event does occur. This is called the probability of the complement, Ac. P(Ac) = 1 – P(A)
An urn has 10 marbles in it. 4 are red, 3 are blue, 2 are green and 1 is purple. P(red) = ? P(not green) = ? P(green or blue) = ? P(striped) = ?
Chapter 15 • The union (U) of two events: when either one or the other, or both events occur. • The intersection (∩) of two events: when both event A and event B occur.
Conditional Probability: P(B|A) • This is read as “the probability of B given A. It is the conditional probability that event B occurs given that event A occurs.
There are two rules of probability that are given on the AP Statistics formula sheet. The first is: • General Addition Rule for Union of Two Events For any two events A and B, P(A U B) = P(A) + P(B) – P(A ∩ B)
General Multiplication Rule The second formula for probability on the AP Stat formula chart is for conditional probability: When P(A) > 0, P(A|B) = P(A ∩ B) P(B)
Independent Events – formally Two events A and B are independent if P(B|A) = P(B) • In other words, the probability of event B is not affected by the event A. This fits with the informal definition of independence but now gives us a way to show two events are independent mathematically. This formula is not on the AP Stat formula chart.
Making Tree Diagrams… Making Tables….
Example1: page 341 #24 • The American Red Cross says that about 45% of the U.S. Population has Type O blood, 40% Type A, 11% Type B, and rest Type AB. • A) Someone volunteers to give blood. What is the probability that this donor • 1. has Type AB blood? • 2. has Type A or Type B? • 3. is not Type O?
P(AB) = .04 or 4% • P(A or B) = .51 or 51% • P(not O) = .55 or 55%
Example 1 continued • The American Red Cross says that about 45% of the U.S. Population has Type O blood, 40% Type A, 11% Type B, and rest Type AB. • B) Among four potential donors, what is the probability that • 1. All are Type O? • 2. no one is Type AB? • 3. they are not all Type A? • 4. at least one person is Type B?
P(all 4 O) = (.45)(.45)(.45)(.45) = .041 • P(no AB) = (.96)(.96)(.96)(.96) = .849 • P(all not A) = 1- P(all A) = 1 – (.4)^4 = .9744 • P(at least 1 B) = 1 – P(no B) = 1 – (.89)^4 = .3726
Example 2: Birthday Gifts • In a book titles, Birthday Gifts, Brier and Rogers found that 71% of married men will get flowers or buy jewelry for their wives on their birthdays. 58% of married men will buy jewelry and 39% will buy flowers for their wives. What is the probability that a married man will buy flowers and jewelry for his wife’s birthday? (Hint: Use a Venn)
Example 3: Oh, Say can you see? • In the book Chances: Risk and Odds in Everyday Life, James Burke says that 56% of the general population wears eyeglasses, while only 3.6% wears contacts. He also says that of those who wear glasses, 55.4% are women. Of those who wear contacts, 36.9% are men. Assume that no one wears both glasses and contacts. For the next person you encounter at random, what is the probability that this person is: • A) not wearing contacts • B) a woman wearing glasses • C) a man wearing contacts
Example 4: Brown Hair/ Brown Eyes • In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town. • 1. If he or she has brown hair, what is the probability that he or she also has brown eyes? • 2. If he or she has brown eyes, what is the probability that he of she does not have brown hair? • 3. What is the probability that he or she has neither brown hair or brown eyes?
Example 5: What does your GPA say? • This example does not fit on one slide, so we will have to use the document camera!!!
