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CH-6 PPT

# CH-6 PPT - Randomness and Probability Randomness and...

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Randomness and Probability

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Randomness and Probability models Probability and Randomness Sample spaces Probability properties Assigning probabilities: equally likely
Probability 1. What is the probability that a flipped coin comes up heads? 2. What are the odds that Microsoft stock price will go up tomorrow? What is the chance of

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A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Randomness and probability The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions.
Coin toss The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip). First series of tosses Second series The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

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Probability models describe, mathematically, the outcome of random processes. They consist of two parts: 1) S = Sample Space : This is a set, or list, of all possible outcomes of a random process. An event is a subset of the sample space. 2) A probability for each possible event in the sample space S . Probability models Example: Probability Model for a Coin Toss : S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5
Sample Space Every possible outcome AKA Universe or population S or Ω A S outcomes

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Probability P (A) = Size of the Event A Size of the Sample Space S General Purpose Definition
Simple Case If outcomes are equally likely P (A) = # outcomes in A Total # outcomes P (Heads) = P (Draw a King) = ½ 4/52 = 1/13

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Flip a Coin Three Times Outcomes HHH HHT HTH HTT THH THT TTH TTT 1. P (HHH) = 1/8 = 0.125 2. P (Two Heads) = 3. P (At least 2 Heads) = 3/8 = 0.375 4/8 = ½ = 0.5
Roll Two Dice Outcomes: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) P (Sum =2) = 1/36 = 0.0278 P (Sum=9) = 4/36 = 1/9 = 0.111 P (Sum=7) = 6/36 = 1/6 = 0.167

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The gambling industry relies on probability distributions to calculate the odds of winning. The rewards are then fixed precisely so that, on average, players lose and the house wins.
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