1. Probability

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

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

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(ADJ  N) = P(2000) P(12000) P(N  ADJ) = P(2000) P(3500) = 0.1666 = 0.5714

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

10. Probability distribution T H HHHTTHTT

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

12. Probability distribution HH HT TH TT 013013 Sample spaceRandom variable

13. Probability distribution Cumulative outcome 0 = 1  1 = 2  2 = 1 

14. Probability distribution Cumulative outcomeProbability 0 = 1  1 = 2  2 = 1  0.25 0.50 0.25  P(x) = 1

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. T H HH HTTHTT HHHHHTHTHHTTTHHTHTTTHTTT

19. Sample space:HHHTTT HHTTTH HTHTHT THHHTT 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 = 0.125

20. 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 Example 2

22. 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?

23. Samples 42 : 58 Population 4 : 4?

24. Normal distribution

25. 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).

26. Skewed distribution

27. Bimodal distribution

28. Skewed distribution

29. Random distribution

30. Normal distribution

31. Example Boys MLUGirls MLU 2.7 2.9 2.6 2.3 3.2 2.9 2.6 3.2 2.9 3.0 3.4 3.2 3.3 2.9 2.74 3.12

32. Example Inspection of data: 1.Frequency – ordinal –interval 2.Normally distributed – not normally distributed

33. Boys 2.8 Girls 3.3 Boys Girls