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Randomness, Uncertainty, & Probability

Randomness, Uncertainty, & Probability. Probability. The formal study of the laws of chance Examples of probability statements are everywhere: There is a 60% chance of rain today The chance of me winning the lottery is 1 in a million

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Randomness, Uncertainty, & Probability

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  1. Randomness, Uncertainty, & Probability

  2. Probability • The formal study of the laws of chance • Examples of probability statements are everywhere: • There is a 60% chance of rain today • The chance of me winning the lottery is 1 in a million • There is a 50-50 chance of observing a head when a fair coin is tossed • There is a 1 in 6 chance in rolling a 6 with a single die.

  3. Randomness • Unpredictability • Cannot predict with any real certainty

  4. Uncertainty • Outcomes are typically known • Which outcome will occur can be predicted w/ some certainty, but not 100% • I think I can, Maybe I can, I should be able to…

  5. Probability • Outcomes are typically known • Which outcome will occur can be predicted w/ some certainty • Measure used to quantify the amount of “doubt” • Replaces the “I think; Maybe” statements with a specific value of certainty. • I am 99% sure I can make the light • I am 0.60 sure I can hit the ball • I am 0.27 sure I will pass this class 

  6. Probability • Theoretically takes place in a Sample Space • All of the possible outcomes listed in a set of brackets • Ex: When a child is born, the sample space would be { B, G} • In a two child family, the sample space would be { BB, BG, GB, GG} • Can use matrix or tree graph to explore more complex outcomes

  7. Sample Space (Graphically Displayed) First Child Outcomes: Boy (B) Girl (G) Second Child Outcomes: Girl (G) Boy (B)

  8. Sample Space (Graphically Displayed) And the 2nd child is If the 1st child is Sample Space BB BG GB GG Boy (B) Boy (B) Girl (G) Possible Outcomes Boy (B) Girl (G) Girl (G)

  9. Probability • Theoretically each outcome of a Sample Space is an Event • If only interested in the outcome BG, then, BG would be the event of interest

  10. Classical Probability • We will denote the probability of an event E as P(E) • Roll a die: P(2), P(5) • Deck of cards: P(Ace), P(Spade) • Roll 2 dice: P(2,6) • The previous formula will then be denoted as: • Where n(E) is the number of events E • And n(S) is the total number of events in the sample space

  11. Classical Probability • Ex: If a two-child family is selected at random, what is the probability of there being one boy and one girl with the girl born 1st? • Simple Event = GB = 1 Sample Space = {BB, BG, GB, GG} = 4 • So: P(GB) = 1 = 0.25 4 • The probability of there being one boy and 1 girl with the girl born 1st in a two-child family is 0.25 or 25 %.

  12. Classical Probability Rolling Two Dice Sample Space • What is P(2,5)? • What is the P(Sum of Dice is a 7)? There is a .17 or 17% chance of rolling a pair of dice and the sum of the two being 7.

  13. Empirical Probability • AKA: Relative Frequency • The probability of an event occurring is the proportion of times the event occurs over a given number of trials • Trials must be repeated exactly (norm, control) • So: P(E) = frequency for the class number of trials

  14. Empirical Probability • Ex: For the 1st 43 presidents of the US, 26 were lawyers. What is the probability of randomly selecting a lawyer from the entire group of 43? • (EA) = the event of a president being a lawyer • So: P(E) = f 26 = 0.61 n 43 • The probability of randomly selecting a lawyer from the past elected presidents is 0.61 or 61 %.

  15. Empirical Probability • Ex: During the flu season, a health clinic observed that on one day, 12 of 60 students examined had strep throat, whereas one week later 18 of 75 students examined had strep throat. What is the relative frequency for each given day? • (EA) = Relative Frequency Day 1 • (EB) = Relative Frequency Day 2 • So: P(EA) = 12 = 0.20 60 • P(EB) = 18 = 0.24 75 • If the data were collected over multiple day, the clinic could average the relative frequencies (generalize) to make one general statement: • During flu season, a student who is examined will have strep throat 0.22 or 22% of the time.

  16. Law of Large Numbers • The empirical probability for an event will change from trial to trial • When repeated a large number of times, the relative frequency approaches the classical probability for the event.

  17. Subjective Probability • Probability measure of belief • Depends on life experiences of the subject. • Sample must be explicitly defined • Cannot be generalized outside the description of the sample • EX: What are the chances that I will have an umbrella when it rains? • Ex: What is the probability patients in cancer remission believe they will live beyond the next 15 years?

  18. General Probability Rules • Law 1: If the probability of an event is 1.00 or 100%, then the event MUST occur. • Law 2: If the probability of an event is 0.00 or 0%, then the event MUST NEVER occur. • Law 3: The probability of any event must assume a value between 0.00 and 1.00. • Law 4: The sum of the probabilities of all the simple events in a sample space must equal 1. • If there are 8 simple events, each event has a 1/8 chance of occurring. • If we sum these probabilities, we have 8 X 1/8 = 1.00

  19. General Probability Rules • The closer the probability to 1.00, the more likely it is to occur. • The closer the probability to 0.00, the less likely it is to occur. • Compound Event • An event that is defined by combining two or more events. • Let: A = students owning a laptop B = students owning an iPhone C = students owning a laptop & an iPhone To Discuss the event C, the researcher looks at the commonalities of both A and B.

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