probability. p(6) = 1/6 = 0.1666 sample space:1,2,3,4,5,6
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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