Probability

Probability

P(6) = 1/6 = 0.1666

Sample space: 1,2,3,4,5,6

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.

Probability

Law of large numbers

The larger the sample space, the closer the

sample distribution to the theoretical distribution.

Joint Probability

P(5,6) = (0.166) P(0.166) = 0.0277

P(A,B) = P(A) P(B)

Conditional Probability

P(AB ) =P(A B)

P(B)

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?

Conditional Probability

P(ADJN) =P(ADJ N)

P(N)

P(ADJN) =P(2000)

P(12000)

P(NADJ) =P(2000)

P(3500)

= 0.1666

= 0.5714

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

Probability distribution

TH

HH HT TH TT

Probability distribution

0 heads = HH

1 head = HT + TH

2 heads = TT

Probability distribution

HH

HT

TH

TT

0

1

3

Sample space Random variable

Probability distribution

Cumulative outcome

0 = 11 = 22 = 1

Probability distribution

Cumulative outcome Probability

0 = 11 = 22 = 1

0.250.500.25

P(x) = 1

Binomial distribution

• two possible outcomes on each trail

• the outcomes are independent of each other

• the probability ratio is constant across trails

Bernoulli trail:

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.

Example 1

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

TH

HH HT TH TT

HHH HHT HTH HTT THH THT TTH TTT

Sample space: HHH TTTHHT TTHHTH THTTHH HTT

Random variables: 0 Head1 Head2 Heads3 Heads

0 head: 11 head: 32 heads: 33 heads: 1

/ 8 = 0.125/ 8 = 0.375/ 8 = 0.375/ 8 = 0.125

If you toss a coin 8 times what is the probability of obtaining a score of:

0 heads1 head2 heads3 heads4 heads5 heads6 heads7 heads8 heads

Example 2

Probability Distribution

Sample:Tossing a coin a 100 times, yielded 42

heads and 58 tails. Is this a fair coin?Heads: 42Tails: 58

Expected: 50% - 50%

Sample error?

Samples42 : 58

Population4 : 4?

Normal distribution

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).

Skewed distribution

Bimodal distribution

Skewed distribution

Random distribution

Normal distribution

Example

Boys MLU Girls MLU

2.72.92.62.33.22.92.6

3.22.93.03.43.23.32.9

2.74 3.12

Example

Inspection of data:

1. Frequency – ordinal –interval

2. Normally distributed – not normally distributed

Boys 2.8

Girls 3.3

Boys

Girls