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Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

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6.5 Combining Probabilities Find the probability. What is the probability of rolling 2 or a 5 on a number cube? 2/6 or 33.33% A bag contains 32 red marbles, 30 blue marbles, and 18 white marbles. You pick one marble from the bag. Find P (picking blue). 3/8 or 37.5% P (not a red) 3/5 or 60% What is the probability of having a sample with mean age between 35 years and 45 years, given the population mean is 40 years and the standard of deviation is 2.5 years? 95% Using a regulation deck of cards. What is the probability of choosing a Queen of Hearts? 1/52, 0.019, or 1.9%

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6.5 Combining Probabilities Contingency Tables Tree Diagrams Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Full Deck of 52 Cards Red Card Black Card Not an Ace Ace Not an Ace Sample Space 2 24 2 24

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6.5 Combining Probabilities Venn Diagrams Let A = aces Let B = red cards A B A ∩ B = ace and red A U B = ace or red

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6.5 Combining Probabilities PERMUTATIONS = Arrangements (Order matters) Permutations: The number of ways of arranging X objects selected from n objects in order is Example: Your restaurant has five menu choices, and three are selected for daily specials. How many different ways can the specials menu be ordered? Answer: different possibilities

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6.5 Combining Probabilities COMBINATIONS = Grouping (Order does not matter) Combinations: The number of ways of selecting X objects from n objects, irrespective of order, is Example: Your restaurant has five menu choices, and three are selected for daily specials. How many different special combinations are there, ignoring the order in which they are selected? Answer: different possibilities

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6.5 Combining Probabilities Joint Probabilities (AND probabilities) Independent VS. Dependent… Independent events are events where the outcomes of one does not affect the outcomes of another. Dependent events are events where the outcome of one will affect the outcome of another. Independent → flipping a coin Dependent → Drawing two cards after drawing a card

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6.5 Combining Probabilities Independent… AND probability… Considering two independent events A and B that have individual probabilities P(A) and P(B). The probability that A and B occur together is: Concept may be extended for more than 2 events.

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6.5 Combining Probabilities Independent…Example Suppose you have a coin and a spinner with 5 equal sectors, labeled 1 thru 5. What is the probability of spinning an even number AND getting heads?

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6.5 Combining Probabilities Dependent … {Conditional Probability} AND probability… Considering two events A and B. The probability that A and B occur together is: Concept may be extended for more than 2 events.

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6.5 Combining Probabilities Dependent … The game of BINGO involves drawing pieces with a letter and a number on each piece. If we draw at random without replacement. Find the probability of drawing two B pieces in the first two selections, given there are 75 pieces, 15 for each of the letters B, I, N, G, O!

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6.5 Combining Probabilities Either/OR Probability: [Disjunction] NON-OVERLAPPING EVENTS… Two events that can not occur at the same time, the probability that either A or B occurs is Concept may be extended for more than 2 events.

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6.5 Combining Probabilities Either/OR Probability… Example… NON-OVERLAPPING EVENTS… What is the probability of rolling a die and getting a 3, 4, or 7?

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6.5 Combining Probabilities Either/OR Probability: OVERLAPPING EVENTS… When two events are considered either/or, but may occur at the same time, then the probability that A or B occurs is:

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6.5 Combining Probabilities MENWOMEN American26 French48 Either/OR Probability: OVERLAPPING EVENTS… Consider this situation on tourism…Given the table, what is the probability of meeting at random a person who is either a woman or French?

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6.5 Combining Probabilities AND probability Independent events AND probability Dependent events Either/OR probability Non- overlapping events Either/OR probability Overlapping events Summary of Combining Probabilities: QUESTIONS????

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6.5 Combining Probabilities HOMEWORK: pg 274 # 1-27 all

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