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03 Probability

# 03 Probability - University of California Los Angeles...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment: A process that results in one of possible outcomes. The outcomes cannot be predicted with certainty. Examples: Flip a coin, roll a die, roll two dice, draw a card, etc. 2. Sample space of a random experiment: It is the list of all possible outcomes of the random experiment, denoted with S . Examples: a. Flip a coin: S = { H,T } . b. Flip two coins: S = { HH,TT,HT,TH } . c. Roll a die: S = { 1 , 2 , 3 , 4 , 5 , 6 } . d. Draw a card: S = { A ♣ ,A ♠ ,A ♦ ,A ♥ , ···} (all 52 cards). 3. Event: It is the outcome of an experiment, denoted with uppercase let- ter. It is a subset of the sample space. Examples: a. Flip a coin: A = { H } . b. Roll a die: A = { even number } , B = { odd number } , C = { 1 , 2 , 3 } . d. Draw a card: A = { Ace } , C = { Clubs } . 4. Probability of an event A for equally likely outcomes: P ( A ) = number of ways in which A occurs number of ways in which all outcomes occur 1 Examples: a. Draw a card. Let A = { Ace } . Then P ( A ) = 4 52 . b. Roll two dice. There are 6 × 6 possible outcomes. The sum of the two numbers rolled are shown below: 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Let A = { sum=5 } . Then P ( A ) = 4 36 . 4. Basic principle of counting: If an experiment has m outcomes and if for every outcome of this experiment there are n out- comes of another experiment then all together there are m × n outcomes. Examples: a. Roll two dice: 6 × 6 = 36 outcomes. b. Flip two coins: 2 × 2 = 4 outcomes. Similarly: a. Roll three dice: 6 × 6 × 6 = 216 outcomes. b. Flip three coins: 2 × 2 × 2 = 8 outcomes. 2 Union of two events A, B The union of two events A , B , denoted A ∪ B , is a new event. It is defined as the event containing all outcomes in A or B or BOTH. Similarly the union of n events A 1 ,A 2 , ··· ,A n is denoted with A 1 ∪ A 2 ∪ ··· ∪ A n . Key word: OR . Example: Suppose 50 students can be classified by their major and year as follows: Sop. Jun. Sen. Total Econ 10 20 6 36 Math 5 4 5 14 Total 15 24 11 50 A student is selected at random from this group of 50 students. Let A = { student is senior } , and B = { student is math major } . Find P ( A ∪ B ). Intersection of two events A, B The intersection of two events A , B , denoted A ∩ B , is a new event. It is defined as the event containing all outcomes that belong to both A and B . Similarly the intersection of n events A 1 ,A 2 , ··· ,A n is denoted with A 1 ∩ A 2 ∩ ··· ∩ A n . Key word: AND ....
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03 Probability - University of California Los Angeles...

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