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Basic Probability

Basic Probability. xkcd.com. One last point from last Week: The Importance of Replication in Experiments. Probability. Are the things that we observe different from what would be expected by chance?. Why Probability Matters in Statistics.

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Basic Probability

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  1. Basic Probability xkcd.com

  2. One last point from last Week: The Importance of Replication in Experiments

  3. Probability • Are the things that we observe different from what would be expected by chance?

  4. Why Probability Matters in Statistics • Statistics is all about chance. We are never testing fact vs truth. We are testing how likely or unlikely something is due to chance. • This has clear implications for every statistical test we will use. • This has clear implications for sampling.

  5. A human probability problem • Myers and Chen (1996) conducted a study where they tested teenagers who had previously participated in an experiment 12 years earlier when they were young children. The teenagers were presented with two objects, only one of which they had seen 12 years ago. Could the participants remember which one they had seen before? • How do we determine if their choice responses are ‘real’ or just chance?

  6. Such probability problems require two things: • Probability Distribution • What would the distribution be like if it were only due to chance? • Decision Rule • What criteria do we need in order to determine whether an observation is just due to chance or not.

  7. Probability Distributions • The probability of an outcome is the relative frequency that an event can be expected to occur. • The probability distribution is the set of relative frequencies for every possible outcome. Probability distribution of number of correct responses for 4 teenagers when p(correct) = .5

  8. Back to Basics: Probability Concepts • Basic Concepts in Probability • Basic Probability Rules • Connecting Probability to Sampling

  9. Basic Concepts in Elementary Probability • Random Selection • Every possibility has equal chance of being chosen. • Independence • The probability of a response on one trial does not depend on the outcome of any other trials (e.g., an event that occurs does not change the chance of some other event occurring). • Elementary Event • Possible outcomes of a probability experiment • E.g., each coin toss, selecting a participant from a population • Sample Space • The complete set of elementary events • E.g., all coin tosses

  10. Mutually exclusive, exhaustive events • Mutually exclusive events • Two or more events that cannot occur at the same time because one prevents the other from occurring or being true. • Exhaustive events • A set of events that accounts for all of the elementary events in the sample space. • What happens to probability when your sample space is not exhaustive? • Why might this happen?

  11. Basic rules of probability • Multiplication Rule • Example: Probability of rolling a die twice and getting 6 on both rolls? • But what happens if the events are not independent? • Example: probability of selecting a club from a deck of cards, then selecting another club (without replacement)?

  12. Multiplication Rule • When two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability:P(X and Y) = P(X) x P(Y) • When two or more events will happen at the same time, and the events are dependent, then the general rule of multiplication law is used to find the joint probability:P(X and Y) = P(X) x P(Y|X)

  13. Pitfalls of the Multiplication Rule and Basic Probability: The Collins Case “Assuming the prosecution's 1 in 12 million result, what is the probability that somewhere in the Los Angeles area there are at least two couples that have the six characteristics as the witnesses described for the robbers? The justices calculated that probability to be over 40 percent. Hence, it was not at all reasonable, they opined, to conclude that the defendants must be guilty simply because they have the six characteristics in the witnesses' descriptions.” • A black man and a white woman were charged with robbery. • Probabilities of various characteristics were multiplied (being in a yellow car, 1/10; man with mustache ¼; Interracial couple in car 1/1000…and so on)

  14. Addition Rule and Mutually Exclusive Events • The addition rule • Example: Rolling die and getting a 4 or a 6? • When two or more events will happen at the same time, and the events aremutually exclusive, then:P(X or Y) = P(X) + P(Y)

  15. Addition Rule and Non-Mutually Exclusive Events • When two or more events will happen at the same time, and the events are not mutually exclusive, then:P(X or Y) = P(X) + P(Y) - P(X and Y)For example, what is the probability that a card chosen at random from a deck of cards will either be a king or a heart? • P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77%

  16. Special Type of Probability: Conditional Probability • Probability of an event occurring given that another event has occurred. • 3 Doors Problem • Basics of the game: Three doors, you pick one. The host of the game then opens one of the other two doors that is *not* the prize. You can then stay with your original choice, or switch. • Another Version with Auto Simulation http://mste.illinois.edu/reese/monty/MontyGame5.html

  17. Fun with probability and sampling:Random numbers – Pick a number between 1-10 and write it down on a piece of paper.

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