AP Statistics Chapter 5 - Probability Notes PDF

AP Statistics Chapter 5 – Probability: What are the Chances? 5.1: Randomness, Probability and Simulation Probability The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions....

AP Statistics Chapter 5 – Probability: What are the Chances? 5.1: Randomness, Probability and Simulation Probability The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Performing of a Simulation – The 4-Step Process 1. State: Ask a question of interest about some chance process. 2. Plan: Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition. 3. Do: Perform many repetitions of the simulation. 4. Conclude: Use the results of your simulation to answer the question of interest. 5.2: Probability Rules Sample Space The sample space S of a chance process is the set of all possible outcomes. Probability Models Descriptions of chance behavior contain two parts: A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome. For example: When a fair 6-sided die is rolled, the Sample Space is S = {1, 2, 3, ,4,5, 6}. The probability for a fair die would include the probabilities of these outcomes, which are all the same. Outcome 1 2 3 4 5 6 Probability 1/6 1/6 1/6 1/6 1/6 1/6 Event An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. For example: For the probability model above we might define event A = die roll is odd. The elements of the sample space S that fits this event are {1, 3, 5}. The probability of the event A, written as P(A) is the 3/6 or ½. So we would write P(A) = 0.5, in decimal form. AP Statistics – Chapter 5 Notes: Probability: What are the Chances? Page 1 of 3 The Basic Rules of Probability For any event A, 0 ≤ P(A) ≤ 1. If S is the sample space in a probability model, P(S) = 1. In the case of equally likely outcomes, number of outcomes corresponding to event A P( A)  total number of outcomes in sample space Complement rule: P(A ) = 1 – P(A) C Addition rule for mutually exclusive events: If A and B are mutually exclusive, P(A or B) = P(A) + P(B). Also be familiar with the notation: 𝑷(𝑨 ∪ 𝑩). Mutually Exclusive Events Two events A and B are mutually exclusive (or disjoint) if they have no outcomes in common and so can never occur together—that is, if P(A and B ) = 0. Alternate notation: 𝑷(𝑨 ∩ 𝑩). For example: Using a deck of playing cards and drawing a card at random, the events A = card is a King, and B = card is a Queen are mutually exclusive because a single card cannot be both a King and a Queen. Thus we can calculate the probability of A or B as the sum of their individual probabilities - P(A or B) = P(A) + P(B). General Addition Rule If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) – P(A and B) Venn Diagrams and Probability The complement Ac contains exactly the The events A and B are mutually exclusive outcomes that are not in A. (disjoint) because they do not overlap. That is, they have no outcomes in common. The intersection of events A and B (A ∩ B) The union of events A and B (A ∪ B) is the is the set of all outcomes in both events A set of all outcomes in either event A or B. and B. AP Statistics – Chapter 5 Notes: Probability: What are the Chances? Page 2 of 3 5.3: Conditional Probability and Independence Conditional Probability The probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A). The symbol “” is read as “given that,” so we read P(B | A) as the probability that B occurs given that A has already occurred. Calculating Conditional Probability To find the conditional probability P(A | B), use the formula P( A  B) P( A | B)  P( B) The conditional probability P(B | A) is given by P( B  A) P( B | A)  P( A) The General Multiplication Rule The probability that events A and B both occur can be found using the general multiplication rule P(A ∩ B) = P(A) P(B | A), where P(B | A) is the conditional probability that event B occurs given that event A has already occurred. Conditional Probability and Independence Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B). The Multiplication Rule for Independent Events If A and B are independent events, then the probability that A and B both occur is P(A ∩ B) = P(A) P(B) AP Statistics – Chapter 5 Notes: Probability: What are the Chances? Page 3 of 3