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Chapter 7 Probability
In this section, we go over some simple concepts from probability theory. We integrate these with ideas from formal language theory in the next chapter.
Probability theory is a conceptual and mathematical treatment of likelihood. The basic idea is that, in some domain, events may not occur with certainty and probability allows us to treat such events in a formal and precise fashion.
We might think that linguistic events are all certain. That is, theories of language are not concerned with judgments or intuitions as events that occur with something less than 100% certainty. For example, we treat the grammaticality or ungrammaticality of any particular sentence as being cer- tain. Thus John loves Mary is completely grammatical and loves John mary is completely ungrammatical.
Less clear judgments are usually ascribed to performance factors. For example, the center-embedded constructions we considered in the previous chapter exhibit gradient acceptability, rather than grammaticality, at least on orthodox assumptions. Acceptability differences are taken to be a con- sequence of performance, while grammaticality differences are taken to be a consequence of grammar.
There are, however, a number of places where probability does play a role in language. Consider first typological distribution, the range of patterns that occur in the languages we know about. There are presumably an infinite number of possible languages, yet the set of extant described languages is\
In this section, we define the notion of a probability distribution, the range of values that might occur in some space of outcomes. To do this, we need the notion of a random variable. Let’s define a random variable as follows.
Definition 20 (Random variable) A random variable is a function that assigns to each outcome in a sample space a unique number. Those numbers exhibit a probability distribution.
Let’s consider an example. The sample space is the number of heads we might throw in three consecutive throws of a coin.
(7.16)
throw number of heads (H,H,H) 3
(H,T,H) 2 (H,H,T) 2 (H,T,T) 1 (T,H,H) 2 (T,T,H) 1 (T,H,T) 1 (T,T,T) 0
There is one way to get three heads, three ways to get two heads, three ways to get one heads, and one way to get no heads. The probability of throwing a head on one throw is p(H) = .5. The probability of throwing a tails is then p(T) = 1 − p(H) = .5. Call the random variable X. The probability of throwing some number of heads can be computed as follows for each combination.
(7.17) p(X =3)=1·p(H)3 ·p(T)0 =1·(.5)3 ·(.5)0 =1·.125·1=.125 p(X =2)=3·p(H)2 ·p(T)1 =3·(.5)2 ·(.5)1 =3·.25·.5=.375 p(X =1)=3·p(H)1 ·p(T)2 =3·(.5)1 ·(.5)2 =3·.5·.25=.375 p(X =0)=1·p(H)0 ·p(T)3 =1·(.5)0 ·(.5)3 =1·1·.125=.125
The logic here is that you multiply together the number of possible combi- nations, the chances of success, raised to the same number, and the chances of failure, raised to the difference.
This is a binomial random variable. It describes the distribution of “suc- cesses” across some number of trials. If we have n trials and are interested in the likelihood of r successes, where the chance of success is p and the chance of failure is q = 1−p, then we have:
n! n
(7.18) p(X = r) = · prqn−r = · prqn−r
This chapter has introduced the basics of probability theory. We began with a general characterization of the notion and proceeded to a mathematical one.
3n The expression exp(n) is equivalent to e .
4 The standard deviation is defined as: (x − μ)2 , where x is each value in the distribution.
density
0.0 0.1 0.2 0.3 0.4
We then considered some basic ideas of combinatorics, how to calculate the number of ways elements can be ordered or chosen.
We presented some basic laws of probability theory, including Bayes’ Law, and we defined the notions of joint probability and conditional probability.
Finally, we briefly discussed the binomial and normal distributions.
7.8 Exercises
1. What is the probability of throwing a 3 with one throw of one die?
2. What is the probability of throwing a 3 and then another 3 with two throws of a single die?
3. What is the probability of throwing a 3 and then a 6 with two throws of a single die?
4. What is the probability of throwing anything but a 3 and another 3 with two throws of a single die?
5. What is the probability of not throwing a 3 at all in two throws of a single die?
6. What is the probability of throwing at least one 3 with two throws of a single die?
7. What is the probability of drawing a jack in one draw from a full deck of playing cards?
8. What is the probability of drawing four aces in four draws from a full deck of playing cards?
9. What is the probability of not drawing any aces in five draws from a full deck of playing cards?