Statistics Tutorial: 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:
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Two events are mutually
exclusive if they have no
sample points
in common.
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The probability that Event A occurs, given that Event B has occurred, is called
a conditional probability.
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The conditional probability of A, given B, is denoted by the symbol P(A|B).
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The probability that event A will not occur is denoted by P(A').
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 Tools
menu item, which appears in the header of every Stat Trek web page.
Rule of Subtraction
In the previous lesson, we learned two important properties of probability:
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The probability of a sample point ranges from 0 to 1.
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The sum of probabilities of all the sample points in a
sample space 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')
Rule of Multiplication
The rule of multiplication applies to the following situation. We have two
events from the same sample space, and we want to know the probability that
both events occur.
Rule of Multiplication If events A and
B come from the same sample space, the probability that both A and B occur is
equal to the probability the event A occurs times the probability that B
occurs, given that A has occurred.
P(A ∩ B) = P(A) P(B|A)
Example 1
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:
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In the beginning, there are 10 marbles in the urn, 4 of which are black.
Therefore, P(A) = 4/10.
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After the first selection, there are 9 marbles in the urn, 3 of which are
black. Therefore, P(B|A) = 3/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15
Example 2
Suppose we repeat the experiment of Example 1; but this time we select marbles with
replacement. That is, we select one marble, note its color, and then
replace it in the urn before making the second selection. When we select with
replacement, 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:
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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(B|A) = 4/10.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 4/25
Rule of Addition
The rule of addition applies to the following situation. We have two events from
the same sample space, and we want to know the probability that either event
occurs.
Rule of Addition If events A and B come
from the same sample space, the probability that event A and/or event B occur
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 1
A student goes to the library. The probability that she checks out (a) a work
of fiction is 0.40, (b) a work of non-fiction is 0.30, , and (c) both fiction
and non-fiction is 0.20. What is the probability that the student checks out a
work of fiction, non-fiction, or both?
Solution: Let F = the event that the student checks out fiction; and let
N = the event that the student checks out non-fiction. 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
Example 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?
Solution: 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:
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There are 52 cards in the deck.
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There are 13 spades, so P(S) = 13/52.
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There are 4 aces, so P(A) = 4/52.
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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
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