Statistical Methods for Psychology

134 Chapter 5 Basic Concepts of Probability

Exercises

5.1 Give an example of an analytic, a relative-frequency, and a subjective view of probability.

5.2 Assume that you have bought a ticket for the local fire department lottery and that your

brother has bought two tickets. You have just read that 1000 tickets have been sold.

event can have more than two possible outcomes—a roll of a die has six possible out-

comes; a maze might present three choices (right, left, and center); political opinions could

be classified as For, Against, or Undecided. In these situations, we must invoke the more

general multinomial distribution.

If we define the probability of each of kevents (categories) as and wish to

calculate the probability of exactly outcomes of , outcomes of , ...,

outcomes of , this probability is given by

where Nhas the same meaning as in the binomial. Note that when k 5 2 this is in fact the

binomial distribution, where and.

As a brief illustration, suppose we had a die with two black sides, three red sides, and

one white side. If we roll this die, the probability of a black side coming up is 2/6 5 .333,

the probability of a red is 3/6 5 .500, and the probability of a white is 1/6 5 .167. If we

roll the die 10 times, what is the probability of obtaining exactly four blacks, five reds, and

one white? This probability is given as

At this point, this is all we will say about the multinomial. It will appear again in

Chapter 6, when we discuss chi-square, and forms the basis for some of the other tests you

are likely to run into in the future.

=.081

= 1260 (.333)^4 (.500)^5 (.167)^1 =1260 (.000064)

p(4, 5, 1)=

10!

4!5!1!

(.333)^4 (.500)^5 (.167)^1

p 2 = 12 p 1 X 2 =N 2 X 1

p(X 1 , X 2 ,... , Xk)=

N!

X 1 !X 2 !ÁXk!

pX 11 pX 22 ÁpXkk

eventk

X 1 event 1 X 2 event 2 Xk

p 1 , p 2 ,... , pk

Key Terms

Analytic view (5.1)

Frequentist view (5.1)

Sample with replacement (5.1)

Subjective probability (5.1)

Event (5.2)

Independent events (5.2)

Mutually exclusive (5.2)

Exhaustive (5.2)

Additive law of probability (5.2)

Multiplicative law of probability (5.2)

Sample without replacement (5.2)

Joint probability (5.2)

Conditional probability (5.2)

Unconditional probability (5.2)

Density (5.5)

Combinatorics (5.6)

Permutation (5.6)

Factorial (5.6)

Combinations (5.6)

Bayes’ Theorem (5.7)

Prior probability (5.7)

Posterior probability (5.7)

Bayesian statistics (5.7)

Binomial distribution (5.8)

Bernoulli trial (5.8)

Success (5.8)

Failure (5.8)

Sign test (5.9)

Multinomial distribution (5.10)