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Sets, Combinatorics, and Probability
example 65
What is the probability of
a. getting “heads” when a coin is tossed?
b. getting a 3 with a roll of a die?
c. drawing either the ace of clubs or the queen of diamonds from a standard
deck of cards?
For (a), tossing a coin results in 2 different possible results, heads or tails, but only
the heads result is of interest. The probability of getting heads is “one out of two”
or 1/2. For (b), the roll of a standard 6-sided die has 6 possible results—any of the
numbers 1, 2, 3, 4, 5, or 6 might come up. The 3 is exactly one of these possibilities, so the probability of rolling a 3 is 1/6. For (c), because a standard card deck
contains 52 cards, the action of drawing one card has 52 possible results. Two of
these are successful results (ace of clubs or queen of diamonds), so the probability
of succeeding in this task is 2/52 or 1/26.
On closer examination, our intuitive answers rely on certain assumptions. We
assume that the coin is a “balanced” coin, equally likely to come up heads or tails.
We assume that the die is not “loaded” and that the deck of cards is not “stacked.”
In other words, we assume that each of the possible outcomes is equally likely;
otherwise our probability of success would be different. (Loaded dice and stacked
decks are the motivation for many Western movie gunfights.)
The set of all possible outcomes of an action is called the sample space S of
the action. Any subset of the sample space is called an event. If S is a finite set
of equally likely outcomes, then the probability P(E) of event E is defined to be
P(E) =
0 E 0
0 S 0
(Remember that 0 A 0 denotes the size of a finite set A.)
example 66
Two coins, A and B, are tossed at the same time; each coin is a fair coin, equally likely to come up heads (H) or tails (T). The sample space of this action is
S = {HH, HT, TH, TT}. Here HT denotes that A comes up heads and B comes up
tails, while TH denotes that A comes up tails and B comes up heads; these are two
different outcomes. Let the event E be the set {HH}. The probability of E—that is,
the probability of having both coins come up heads
2
—is
P(E ) =
0 E 0
0 S 0
=
0 5HH6 0
0 5HH, HT, TH, TT6 0
=
1
4
= 0.25
2 An event, while technically a set, is also used to describe the action whose outcomes are the members of
this set. Here the event E is the set {HH}, but we also speak of the event as the action of tossing two coins
and having both come up heads.
■
PraCtiCe 42 Find the probability of drawing an ace from a standard deck of cards.
Sets, Combinatorics, and Probability
example 65
What is the probability of
a. getting “heads” when a coin is tossed?
b. getting a 3 with a roll of a die?
c. drawing either the ace of clubs or the queen of diamonds from a standard
deck of cards?
For (a), tossing a coin results in 2 different possible results, heads or tails, but only
the heads result is of interest. The probability of getting heads is “one out of two”
or 1/2. For (b), the roll of a standard 6-sided die has 6 possible results—any of the
numbers 1, 2, 3, 4, 5, or 6 might come up. The 3 is exactly one of these possibilities, so the probability of rolling a 3 is 1/6. For (c), because a standard card deck
contains 52 cards, the action of drawing one card has 52 possible results. Two of
these are successful results (ace of clubs or queen of diamonds), so the probability
of succeeding in this task is 2/52 or 1/26.
On closer examination, our intuitive answers rely on certain assumptions. We
assume that the coin is a “balanced” coin, equally likely to come up heads or tails.
We assume that the die is not “loaded” and that the deck of cards is not “stacked.”
In other words, we assume that each of the possible outcomes is equally likely;
otherwise our probability of success would be different. (Loaded dice and stacked
decks are the motivation for many Western movie gunfights.)
The set of all possible outcomes of an action is called the sample space S of
the action. Any subset of the sample space is called an event. If S is a finite set
of equally likely outcomes, then the probability P(E) of event E is defined to be
P(E) =
0 E 0
0 S 0
(Remember that 0 A 0 denotes the size of a finite set A.)
example 66
Two coins, A and B, are tossed at the same time; each coin is a fair coin, equally likely to come up heads (H) or tails (T). The sample space of this action is
S = {HH, HT, TH, TT}. Here HT denotes that A comes up heads and B comes up
tails, while TH denotes that A comes up tails and B comes up heads; these are two
different outcomes. Let the event E be the set {HH}. The probability of E—that is,
the probability of having both coins come up heads
2
—is
P(E ) =
0 E 0
0 S 0
=
0 5HH6 0
0 5HH, HT, TH, TT6 0
=
1
4
= 0.25
2 An event, while technically a set, is also used to describe the action whose outcomes are the members of
this set. Here the event E is the set {HH}, but we also speak of the event as the action of tossing two coins
and having both come up heads.
■
PraCtiCe 42 Find the probability of drawing an ace from a standard deck of cards.