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What can we say about probability? It is a measure of likelihood, uncertainty, possibility, … And it is a number, numeric measure

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Set: A set is a collection of distinct objects, considered as an object in its own right Example: A class of stat225 students NBA teams Result of a football game for a team (win, lose, tie) Outcomes of rolling a die ( 1, 2, 3, 4, 5, 6)

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Experiment: A process that generates well-defined outcomes. *** You must know what could possibly happen before performing the experiment. Example: Roll a die: YES ? Throw a stone out of window: NO.

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Sample space Dont get confused with sample and sampling in statistics. A sample space for an experiment is the set of ALL experimental outcomes. Example: Toss a coin: {Head, Tail} Take an exam or quiz: { All the possible grades } Inspection of a manufactured part: {defective, non- defective}

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Some notes on sample space: 1. It could be finite or infinite, i.e., there could be infinitely many possibilities, but you still must know all of them. Example, choosing a point on a segment, there are infinitely many choices but you know it must be on the segment. 2. Not all the sample points in the sample space are equally likely. Example: Getting a royal flush, full house or just a pair in a poker game.

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Event: A collection of sample points. It is an outcome from an experiment that may include one or more sample points. Examples: Toss a coin and get a head. Roll a die and get a 5. Toss a coin twice and get two heads. Roll a die and get an even number.

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Complement of an event A: Notation: A c Also an event Includes ALL sample points that are not in A. Example: A: Roll a die and get an even number, A c : Roll a die and get an odd number.

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We use P(A) to represent the probability that one event occurs. Clearly: P(A) + P(A c )=1. A and A c are called mutually exclusive, which means any sample point call fall in either A only or A c only, but not both.

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Some of tossed a coin twice. 1. What is the sample space: { HH, HT, TH, TT } 2. Let A = { toss the coin twice and get two heads}, then P(A)=? 3. Let B = { toss the coin twice and get at least one head}, then P(B)=? 4. What is B c, and P(B c )=?

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We have 20 fruits in a box, 10 apples, 6 pears and 4 peaches. If A={pick a fruit from the box and get an apple}, then P(A)=? If B={pick a fruit from the box and it is NOT an apple}, then P(B)=?

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50 balls are put in one box, 25 white, 15 red and 10 green. A={ pick a ball and it is green }, P{A}=? B={pick a ball and it is colored}, P{B}=?

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Venn diagram: Illustrates the concept of complement. Somewhat like a crosstabulation.

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# 1. A class of 30 students took 2 midterms during a semester. 22 of them passed the first one and 26 of them passed the second one. If 3 students failed both, find the number of students who passed the each of the two midterms and failed the other. Answer: 21 passed both, 1 passed the first but failed the second, 5 passed the second but failed the first

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#2. We have a box and 30 balls, 16 are white and the rest are colored; 9 are plastic and the others are made of rubber. If there are 11 white rubber balls, show the breakdown of balls by color and material. Answer: 5 white plastic; 11 white rubber; 4 colored plastic and 10 colored rubber.

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#3. In a class of 30 students, 20 are male, 15 are white and 5 are black females. Assuming only no students of other race in the class. Find the break down of students by race and gender. Answer: 10 black male; 5 black female; 5 white female and 10 white male.

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In #1, what is the probability of finding a student who passed at least one midterm? 27/30 In #2, pick a ball at random, what is the probability of getting a white plastic? 5/30 In #3, pick a student from the class and what is the probability of getting a white female? 5/30

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Sixth lecture Concepts of Probabilities. Random Experiment Can be repeated (theoretically) an infinite number of times Has a well-defined set of possible.

Sixth lecture Concepts of Probabilities. Random Experiment Can be repeated (theoretically) an infinite number of times Has a well-defined set of possible.

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