Psych. 304 Worksheet #1
Probability Worksheet
This worksheet is intended to act as a brief “tutorial” covering basic concepts in probability
theory. Have a look at the appendix of the Hastie & Dawes textbook before completing this
worksheet. It will be marked primarily on the basis of effort. Answers will be posted on the
course webpage.
Conjunctions and Disjunctions
Consider two propositions (also referred to as events or possibilities), A and B, each of which
may either be true or false. (One example you can keep in mind while completing this
worksheet is that A is “it will rain tomorrow” and B is “it will be windy tomorrow.”) We can
assign probabilities to each, denoted as p(A) and p(B). We can also judge the probability of the
conjunction of the two events, denoted p(A&B), which is the probability that both A and B are
true. Here is a Venn diagram showing one possible relationship between A and B:
neither A nor B
A
B
A&B
Imagine that we know (or have already judged) the values of p(A), p(B), and p(A&B). Given
these values, how would you compute the value of the disjunction p(A or B), the probability that
either A or B (or both) is true?
Mutual Exclusivity
Two propositions, A and B, are said to be mutually exclusive if they cannot both be true at the
same time (i.e., if one’s being true implies that the other is false). In the special case of A and B
being mutually exclusive, what is the probability of their disjunction p(A or B)? Use your
solution for the previous question to answer this one.
Psych. 304 Worksheet #1
Conditional Probability
The probability of A being true assuming that B is true is denoted p(A | B), and is read “the
probability of A given B.” This is called a conditional probability, because it represents our
revised probability of A under the assumption or condition that B is true. Once again assuming
that we know p(A), p(B), and p(A&B), work out the equation we can use to find p(A | B). (Hint:
In the figure above, assuming that B is true amounts to focusing on the circle representing B, and
excluding the other possibilities falling outside the circle.) Then work out the equation for
finding p(B | A). Under what condition are these two “inverse” conditional probabilities equal?
Chaining Principle
Rewrite the formula you worked out for computing p(A | B) from p(B), and p(A&B), this time
solving for p(A&B). The result is what Hastie & Dawes refer to as the chaining principle,
because the conjunction is thought of in terms of a chain of events. This kind of “chain” is
readily extended to conjunctions of three or more events. Can you find a similar equation for
p(A&B&C)?
Independence
Two propositions A and B are said to be independent if finding out whether one is true or false
has no effect on the probability of the other. Under such circumstances, p(A | B) = p(A), that is,
our probability of A is unchanged by the knowledge that B is true. In the special case of A and
B being independent, what happens to the multiplication rule for determining p(A&B)? How are
the concepts of independence and mutual exclusivity related? That is, are all pairs of
independent events also mutually exclusive, or vice versa?
Bayes’ Rule
What is the relationship between p(A | B) and p(B | A)? That is, if you knew the values of p(A),
p(B), and p(A | B), what is the equation you could use to find p(B | A)? Hint: Combine the two
equations you’ve already worked out for p(A | B) and p(B | A). Your answer should give you a
simple form of Bayes’ rule (or theorem), which can be used to revise one’s probability of some
hypothesis (H) being true in light of new data (D). Simply replace A with D and B with H in the
equation you have just worked out.