The Terminology of Probability

Learning Outcomes

  • Understand and use the terminology of probability


Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin twice is an example of an experiment.

A result of an experiment is called an outcome. The sample space of an experiment is the set of all possible outcomes. Three ways to represent a sample space are: to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter
SS
is used to denote the sample space. For example, if you flip one fair coin,
SS
= {
HH
,
TT
} where
HH
= heads and
TT
= tails are the outcomes.

An event is any combination of outcomes. Upper case letters like
AA
and
BB
represent events. For example, if the experiment is to flip one fair coin, event
AA
might be getting at most one head. The probability of an event
AA
is written
PP
(
AA
).

The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values).
PP
(
AA
) =
00
means the event
AA
can never happen.
PP
(
AA
) =
11
means the event
AA
always happens.
PP
(
AA
) =
0.50.5
means the event
AA
is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from
2020
to
2,0002,000
to
20,00020,000
times) the relative frequency of heads approaches
0.50.5
(the probability of heads).

Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face (
1,2,3,4,5,or61, 2, 3, 4, 5, \text{or}\,6
) is as likely to occur as any other face. If you toss a fair coin, a Head (
HH
) and a Tail (
TT
) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event
AA
when all outcomes in the sample space are equally likely
, count the number of outcomes for event
AA
and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is {
HHHH
,
THTH
,
HTHT
,
TTTT
} where
TT
= tails and
HH
= heads. The sample space has four outcomes.
AA
= getting one head. There are two outcomes that meet this condition {
HTHT
,
THTH
}, so
P(A)=24=0.5\displaystyle{P}{({A})}=\frac{{2}}{{4}}={0.5}
.

Suppose you roll one fair six-sided die, with the numbers {
1,2,3,4,5,61, 2, 3, 4, 5, 6
} on its faces. Let event 
EE
= rolling a number that is at least five. There are two outcomes {
5,65, 6
}.
P(E)=26\displaystyle{P}{({E})}=\frac{{2}}{{6}}
as the number of repetitions grows larger and larger.

This important characteristic of probability experiments is known as the law of large numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.)

This video gives more examples of basic probabilities.



It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in
250250
trials, a head was obtained
5656
% of the time and a tail was obtained
4444
% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.


"OR" Event

An outcome is in the event 
AA
OR
BB
if the outcome is in
AA
or is in
BB
or is in both
AA
and
BB
. For example, let
AA
= {
1,2,3,4,51, 2, 3, 4, 5
} and
BB
= {
4,5,6,7,84, 5, 6, 7, 8
}.
AA
OR
BB
= {
1,2,3,4,5,6,7,81, 2, 3, 4, 5, 6, 7, 8
}. Notice that
44
and
55
are NOT listed twice.


"AND" Event

An outcome is in the event 
AA
AND
BB
if the outcome is in both
AA
and
BB
at the same time. For example, let
AA
and
BB
be {
1,2,3,4,51, 2, 3, 4, 5
} and {
4,5,6,7,84, 5, 6, 7, 8
}, respectively. Then
AA
AND
BB
= {
4,54, 5
}.


The complement of event
AA
is denoted
AA'
(read "
AA
prime").
AA'
consists of all outcomes that are NOT in
AA
. Notice that
PP
(
AA
) +
PP
(
AA'
) =
11
. For example, let
SS
= {
1,2,3,4,5,61, 2, 3, 4, 5, 6
} and let
AA
= {
1,2,3,41, 2, 3, 4
}. Then,
A=5,6A'={5, 6}
.
P(A)=46P(A) = \frac{{4}}{{6}}
and
P(A)=26P(A') = \frac{{2}}{{6}}
, and
P(A)+P(A)=46+26=1P(A) +P(A') =\frac{{4}}{{6}}+\frac{{2}}{{6}}={1}
.

The conditional probability of
AA
given
BB
is written
PP
(
AA
|
BB
).
PP
(
AA
|
BB
) is the probability that event
AA
will occur given that the event
BB
has already occurred. A conditional reduces the sample space. We calculate the probability of
AA
from the reduced sample space
BB
. The formula to calculate
PP
(
AA
|
BB
) is
P(AB)=P(A AND B)P(B)\displaystyle{P}{({A}{|}{B})}=\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}
where
PP
(
BB
) is greater than zero.

For example, suppose we toss one fair, six-sided die. The sample space

SS
= {
1,2,3,4,5,61, 2, 3, 4, 5, 6
}. Let
AA
= face is
22
or
33
and
BB
= face is even (
2,4,62, 4, 6
). To calculate
PP
(
AA
|
BB
), we count the number of outcomes
22
or
33
in the sample space
BB
= {
2,4,62, 4, 6
}. Then we divide that by the number of outcomes
BB
(rather than
SS
).

We get the same result by using the formula. Remember that 
SS
has six outcomes.

P(AB)=P(A AND B)P(B)=the number of outcomes that are 2 or 3 and even in S6the number of outcomes that are even in S6=1636=13\displaystyle{P}{({A}{|}{B})}=\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}=\frac{{\frac{{\text{the number of outcomes that are 2 or 3 and even in } {S}}}{{6}}}}{{\frac{{\text{the number of outcomes that are even in } {S}}}{{6}}}}=\frac{{\frac{{1}}{{6}}}}{{\frac{{3}}{{6}}}}=\frac{{1}}{{3}}


Understanding Terminology and Symbols

It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

Example

The sample space 
SS
is the whole numbers starting at one and less than
2020
.

  1. SS
    = _____________________________Let event
    AA
    = the even numbers and event
    BB
    = numbers greater than
    1313
    .
  2. AA
    = _____________________,
    BB
    = _____________________
  3. PP
    (
    AA
    ) = _____________,
    PP
    (
    BB
    ) = ________________
  4. AA
    AND
    BB
    = ____________________,
    AA
    OR
    BB
    = ________________
  5. PP
    (
    AA
    AND
    BB
    ) = _________,
    PP
    (
    AA
    OR
    BB
    ) = _____________
  6. AA'
    = _____________,
    PP
    (
    AA'
    ) = _____________
  7. PP
    (
    AA
    ) +
    PP
    (
    AA'
    ) = ____________
  8. PP
    (
    AA
    |
    BB
    ) = ___________,
    PP
    (
    BB
    |
    AA
    ) = _____________; are the probabilities equal?






Try it

The sample space 
SS
is the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (
1,41, 4
)).

  1. SS
    = _____________________________Let event
    AA
    = the sum is even and event
    BB
    = the first number is prime.
  2. AA
    = _____________________,
    BB
    = _____________________
  3. PP
    (
    AA
    ) = _____________,
    PP
    (
    BB
    ) = ________________
  4. AA
    AND
    BB
    = ____________________,
    AA
    OR
    BB
    = ________________
  5. PP
    (
    AA
    AND
    BB
    ) = _________,
    PP
    (
    AA
    OR
    BB
    ) = _____________
  6. BB'
    = _____________,
    PP
    (
    BB'
    ) = _____________
  7. PP
    (
    AA
    ) +
    PP
    (
    AA
    ) = ____________
  8. PP
    (
    AA
    |
    BB
    ) = ___________,
    PP
    (
    BB
    |
    AA
    ) = _____________; are the probabilities equal?






Example

A fair, six-sided die is rolled. Describe the sample space
SS
, identify each of the following events with a subset of
SS
and compute its probability (an outcome is the number of dots that show up).

  1. Event
    TT
    = the outcome is two.
  2. Event
    AA
    = the outcome is an even number.
  3. Event
    BB
    = the outcome is less than four.
  4. The complement of
    AA
    .
  5. AA
    GIVEN
    BB
  6. BB
    GIVEN
    AA
  7. AA
    AND
    BB
  8. AA
    OR
    BB
  9. AA
    OR
    BB'
  10. Event
    NN
    = the outcome is a prime number.
  11. Event
    II
    = the outcome is seven.






Try it

The table describes the distribution of a random sample
SS
of
100100
individuals, organized by gender and whether they are right- or left-handed.

Right-handed Left-handed
Males
4343
99
Females
4444
44
Let's denote the events 
MM
= the subject is male,
FF
= the subject is female,
RR
= the subject is right-handed,
LL
= the subject is left-handed. Compute the following probabilities:

  1. PP
    (
    MM
    )
  2. PP
    (
    FF
    )
  3. PP
    (
    RR
    )
  4. PP
    (
    LL
    )
  5. PP
    (
    MM
    AND
    RR
    )
  6. PP
    (
    FF
    AND
    LL
    )
  7. PP
    (
    MM
    OR
    FF
    )
  8. PP
    (
    MM
    OR
    RR
    )
  9. PP
    (
    FF
    OR
    LL
    )
  10. PP
    (
    MM'
    )
  11. PP
    (
    RR
    |
    MM
    )
  12. PP
    (
    FF
    |
    LL
    )
  13. PP
    (
    LL
    |
    FF
    )






 

References

"Countries List by Continent." Worldatlas, 2013. Available online at http://www.worldatlas.com/cntycont.htm (accessed May 2, 2013).

Concept Review

In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

Formula Review

AA
and
BB
are events

PP
(
SS
) =
11
where
SS
is the sample space 0 ≤
PP
(
AA
) ≤
11


PP
(
AA
|
BB
)=
P(A AND B)P(B)\displaystyle\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}


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