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Chapter 5-2 The Idea of Probability AP Statistics Classical Definition The probability of an event E, denoted P(E), is: # of outcomes favorable to E P E # of outcomes in the sample space Note: This method for calculating probabilities is ONLY appropriate when the outcomes of an experiment are equally likely. For example, rolling a die or selecting a card from a deck are experiments that have equally likely outcomes. The Relative Approach When it’s difficult to find a probability, sometimes we use a relative approach and an alternate definition for probability: # of times E occurs P E # of trials as the # of trials becomes very large From P(3)? a lopsided-die, how would we find the # of times a 3 is rolled total # of times the die is rolled Forming New Events Two events that have no outcomes in common are said to be disjoint or mutually exclusive. ex: Bobby Fisher and Borris Spassky are playing chess: How many possible outcomes are there? There are three possibilities… Bobby Fisher winns, Borris wins, or there is a draw (neither win). Can Bobby win if Borris wins? Can Borris win if Bobby wins? Can either win when there is a draw? Probability Venn Diagrams and Probabilities: How do the Raiders Perform when it rains? Worksheet Basic Rules of Probability Forming New Events Let A and B denote 2 events: The event not A consists of all experimental outcomes that are not in event A. This event is often called the complement of A. 2. The event A or B consists of all experimental outcomes that are in at least one of the two events, that is, in A, in B, or in both A and B. This event is called the union of events A and B 3. The event A and B consists of all experimental outcomes that are in BOTH of the events A and B. This event is called the intersection of events A and B 1. Examples Find the probability of each of the following: 1. 2. 3. 4. P(rolling a 6) = ? P(rolling an even #) = ? P(drawing the queen of spades) = ? P(drawing any queen) = ? Which of these events are simple events? Here are some events where the outcomes are not equally likely: Letter grade on the next exam: P(A) ≠ P(B) ≠ P(C) ≠ P(D) ≠ P(F) ≠ 1 / 5 If the probabilities are not equally likely, you cannot use the classical definition. The Relative Approach What is the probability of drawing a face card if I have a messed up collection of cards that includes several decks that have been mixed together with many of the cards missing? If I were to draw a card from the scrambled deck, what are the possible outcomes? Are they equally likely?? How can I estimate P(face card)? I can repeatedly draw cards and keep track of the ratio: # of face cards P(face card) total # of cards drawn Forming New Events List all possible outcomes in the following events: A c A B A B A B c A A c