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Probability

Probability. Probability. Sample space: 1,2,3,4,5,6. P(6) = 1/6 = 0.1666. Probability. Probability values range from 0 to 1. Adding all probability of the sample yields 1. The probability that an event A will not occur is 1 minus the probability of A.

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Probability

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  1. Probability

  2. Probability Sample space: 1,2,3,4,5,6 P(6) = 1/6 = 0.1666

  3. Probability • Probability values range from 0 to 1. • Adding all probability of the sample yields 1. • The probability that an event A will not occur is 1 minus the probability of A. • If two events are independent, the probability is the sum of their individual probabilities. • Two events A and B are independent if knowing that the occurrence of A does not change the probability of the occurrence of B.

  4. Probability Law of large numbers The larger the sample space, the closer the sample distribution to the theoretical distribution.

  5. Joint Probability P(A,B) = P(A)  P(B) P(5,6) = (0.166)  P(0.166) = 0.0277

  6. Conditional Probability P(A B) P(AB) = P(B)

  7. Conditional Probability In a corpus including 12.000 nouns and 3.500 adjectives, 2.000 adjectives precede a noun. (1) What is the likelihood that a noun occurs after an adjective? (2) What is the likelihood that an adjective precedes a noun?

  8. Conditional Probability P(ADJ N) P(ADJN) = P(N) P(2000) P(ADJN) = = 0.1666 P(12000) P(2000) = 0.5714 P(NADJ) = P(3500)

  9. Probability 0.4  0.8 = 0.32 0.8 pronominal transitive 0.4 lexical 0.4  0.2 = 0.08 0.2 0.6 pronominal 0.6  0.6 = 0.36 0.6 intransitive lexical 0.6  0.4 = 0.24 0.4 Sum = 1

  10. Probability distribution T H HH HT TH TT

  11. Probability distribution 0 heads = HH 1 head = HT + TH 2 heads = TT

  12. Probability distribution HH HT TH TT 0 1 3 Sample space Random variable

  13. Probability distribution

  14. Probability distribution

  15. Binomial distribution • two possible outcomes on each trail • the outcomes are independent of each other • the probability ratio is constant across trails Bernoulli trail:

  16. Binomial distribution • It is based on categorical / nominal data. • There are exactly two outcomes for each trail. • All trials are independent. • The probability of the outcomes is the same for each trail. • A sequence of Bernoulli trails gives us the binominal distribution.

  17. Example 1 A coin is tossed three times. What is the probability of obtaining two heads?

  18. H T HH HT TH TT HHH HHT HTH HTT THH THT TTH TTT

  19. Sample space: HHH TTT HHT TTH HTH THT THH HTT Random variables: 0 Head 1 Head 2 Heads 3 Heads 0 head: 1 1 head: 3 2 heads: 3 3 heads: 1 / 8 = 0.125 / 8 = 0.375 / 8 = 0.375 / 8 = 0.125

  20. Example 2 If you toss a coin 8 times what is the probability of obtaining a score of: 0 heads 1 head 2 heads 3 heads 4 heads 5 heads 6 heads 7 heads 8 heads

  21. Probability Distribution Sample: Tossing a coin a 100 times , yielded 42 heads and 58 tails. Is this a fair coin? Heads: 42 Tails: 58 Expected: 50% - 50% Sample error?

  22. Population 4 : 4? Samples 42 : 58

  23. Normal distribution

  24. Normal distribution • The center of the curve represents the mean, median, and mode. • The curve is symmetrical around the mean. • The tails meet the x-axis in infinity. • The curve is bell-shaped. • The total under the curve is equal to 1 (by definition).

  25. Skewed distribution

  26. Bimodal distribution

  27. Skewed distribution

  28. Random distribution

  29. Normal distribution

  30. Example

  31. Example Inspection of data: • Frequency – ordinal –interval • Normally distributed – not normally distributed

  32. Boys 2.8 Boys Girls Girls 3.3

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