Rules of Probability
Often, we want to compute the probability of an event from the known
probabilities of other events. This lesson covers some important rules
that simplify those computations.
Definitions and Notation
Before discussing the rules of probability, we state the following definitions:

Two events are mutually
exclusive or disjoint
if they cannot occur at the same time.

The probability that Event A occurs, given that Event B has occurred, is called
a conditional probability. The conditional probability
of Event A, given Event B, is denoted by the symbol P(AB).

The complement of an event is the event not occurring.
The probability that Event A will not occur is denoted by P(A').
 The probability that Events A and B both occur is
the probability of the intersection of A and B.
The probability of the intersection of Events A and B is denoted by
P(A ∩ B). If Events A and B are
mutually exclusive, P(A ∩ B) = 0.
 The probability that Events A or B occur is
the probability of the union of A and B.
The probability of the union of Events A and B is denoted by
P(A ∪ B) .
 If the occurrence of Event A changes the probability of
Event B, then Events A and B are dependent.
On the other hand, if the occurrence of Event A does not change
the probability of Event B, then Events A and B are
independent.
Probability
Calculator
Use the Probability Calculator to compute the probability of
an event from the known probabilities of other events. The Probability
Calculator is free and easy to use. It can be found under the Stat Tools
tab, which appears in the header of every Stat Trek web page.
Rule of Subtraction
In a
previous lesson,
we learned two important properties of probability:

The probability of an event ranges from 0 to 1.

The sum of probabilities of all possible events equals 1.
The rule of subtraction follows directly from these properties.
Rule of Subtraction The probability
that event A will occur is equal to 1 minus the probability that event A will
not
occur.
P(A) = 1  P(A')
Suppose, for example, the probability that Bill will graduate from college
is 0.80. What is the probability that Bill will not graduate from college?
Based on the rule of subtraction, the probability that Bill will not graduate
is 1.00  0.80 or 0.20.
Rule of Multiplication
The rule of multiplication applies to the situation when we want to know
the probability of the intersection of two events; that is, we want to know
the probability that two events (Event A and Event B) both occur.
Rule of Multiplication The
probability that Events A and B both occur is
equal to the probability that Event A occurs times the probability that
Event B occurs, given that A has occurred.
P(A ∩ B) = P(A) P(BA)
Example
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without
replacement from the urn. What is the probability that both of the
marbles are black?
Solution: Let A = the event that the first marble is black; and let B =
the event that the second marble is black. We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black.
Therefore, P(A) = 4/10.

After the first selection, there are 9 marbles in the urn, 3 of which are
black. Therefore, P(BA) = 3/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(BA)
P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15
Rule of Addition
The rule of addition applies to the following situation. We have two events,
and we want to know the probability that either event occurs.
Rule of Addition The probability that
Event A or Event B occurs
is equal to the probability that Event A occurs plus the probability that Event
B occurs minus the probability that both Events A and B occur.
P(A ∪ B) = P(A) + P(B)  P(A ∩ B)
Note: Invoking the fact that P(A
∩ B) = P( A )P( B  A ), the Addition Rule can also be expressed as
P(A ∪ B) = P(A) + P(B)  P(A)P( B  A )
Example
A student goes to the library. The probability that she checks out (a) a work
of fiction is 0.40, (b) a work of nonfiction is 0.30, and (c) both fiction
and nonfiction is 0.20. What is the probability that the student checks out a
work of fiction, nonfiction, or both?
Solution: Let F = the event that the student checks out fiction; and let
N = the event that the student checks out nonfiction. Then, based on the rule
of addition:
P(F ∪ N) = P(F) + P(N)  P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30  0.20 = 0.50
Test Your Understanding
Problem 1
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn
with replacement from the urn. What is the probability that both of the
marbles are black?
(A) 0.16
(B) 0.32
(C) 0.36
(D) 0.40
(E) 0.60
Solution
The correct answer is A. Let A = the event that the first marble is black;
and let B =
the event that the second marble is black. We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black.
Therefore, P(A) = 4/10.

After the first selection, we replace the selected marble; so there are still
10 marbles in the urn, 4 of which are black. Therefore, P(BA) = 4/10.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(BA)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 0.16
Problem 2
A card is drawn randomly from a deck of ordinary playing cards. You win $10 if
the card is a spade or an ace. What is the probability that you will win the
game?
(A) 1/13
(B) 13/52
(C) 4/13
(D) 17/52
(E) None of the above.
Solution
The correct answer is C.
Let S = the event that the card is a spade; and let A = the
event that the card is an ace. We know the following:

There are 52 cards in the deck.

There are 13 spades, so P(S) = 13/52.

There are 4 aces, so P(A) = 4/52.

There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.
Therefore, based on the rule of addition:
P(S ∪ A) = P(S) + P(A)  P(S ∩ A)
P(S ∪ A) = 13/52 + 4/52  1/52 = 16/52 = 4/13