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4.1: Probability Experiments and Sample Spaces

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    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 S is used to denote the sample space. For example, if you flip one fair coin, \(S = \{\text{H, T}\}\) where \(\text{H} =\) heads and \(\text{T} =\) tails are the outcomes.

    An event is any combination of outcomes. Upper case letters like \(\text{A}\) and \(\text{B}\) represent events. For example, if the experiment is to flip one fair coin, event \(\text{A}\) might be getting at most one head. The probability of an event \(\text{A}\) is written \(P(\text{A})\).

    Definition: Probability

    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).

    • \(P(\text{A}) = 0\) means the event \(\text{A}\) can never happen.
    • \(P(\text{A}) = 1\) means the event \(\text{A}\) always happens.
    • \(P(\text{A}) = 0.5\) means the event \(\text{A}\) is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches 0.5 (the probability of heads).

     

    The "OR" Event

    An outcome is in the event \(\text{A OR B}\) if the outcome is in \(\text{A}\) or is in \(\text{B}\) or is in both \(\text{A}\) and \(\text{B}\). For example, let \(\text{A} = \{1, 2, 3, 4, 5\}\) and \(\text{B} = \{4, 5, 6, 7, 8\}\). \(\text{A OR B} = \{1, 2, 3, 4, 5, 6, 7, 8\}\). Notice that 4 and 5 are NOT listed twice.

    The "AND" Event

    An outcome is in the event \(\text{A AND B}\) if the outcome is in both \(\text{A}\) and \(\text{B}\) at the same time. For example, let \(\text{A}\) and \(\text{B}\) be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then \(\text{A AND B} = {4, 5}\).

    The complement of event \(\text{A}\) is denoted \(\text{A'}\) (read "A prime"). \(\text{A'}\) consists of all outcomes that are NOT in \(\text{A}\). Notice that

    \[P(\text{A}) + P(\text{A′}) = 1. \nonumber\]

    For example, let \(\text{S} = \{1, 2, 3, 4, 5, 6\}\) and let \(\text{A} = {1, 2, 3, 4}\). Then, \(\text{A′} = {5, 6}\) and \(P(A) = \frac{4}{6}\), \(P(\text{A′}) = \frac{2}{6}\), and

    \[P(\text{A}) + P(\text{A′}) = \frac{4}{6} + \frac{2}{6} = 1. \nonumber\]

    The conditional probability of \(\text{A}\) given \(\text{B}\) is written \(P(\text{A|B})\). \(P(\text{A|B})\) is the probability that event \(\text{A}\) will occur given that the event \(\text{B}\) has already occurred. A conditional reduces the sample space. We calculate the probability of \(\text{A}\) from the reduced sample space \(\text{B}\). The formula to calculate \(P(\text{A|B})\) is

    \[P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}} \nonumber\]

    where \(P(\text{B})\) is greater than zero.

    For example, suppose we toss one fair, six-sided die. The sample space \(\text{S} = \{1, 2, 3, 4, 5, 6\}\). Let \(\text{A} =\) face is 2 or 3 and \(\text{B} =\) face is even (2, 4, 6). To calculate \(P(\text{A|B})\), we count the number of outcomes 2 or 3 in the sample space \(\text{B} = \{2, 4, 6\}\). Then we divide that by the number of outcomes \(\text{B}\) (rather than \(\text{S}\)).

    We get the same result by using the formula. Remember that \(\text{S}\) has six outcomes.

    \[ \begin{align*} P(\text{A|B}) &= \dfrac{ \text{ P(A AND B) } } {P(\text{B})} \\[4pt] &= \dfrac{\dfrac{\text{the number of outcomes that are 2 or 3 and even in S}}{6}}{\dfrac{\text{the number of outcomes that are even in S}}{6}} \\[4pt] &= \dfrac{\frac{1}{6}}{\frac{3}{6}} = \dfrac{1}{3} \end{align*}\]

    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 \(\PageIndex{1}\)

    The sample space \(S\) is the whole numbers starting at one and less than 20.

    1. \(S =\) _____________________________

      Let event \(A =\) the even numbers and event \(B =\) numbers greater than 13.

    2. \(A =\) _____________________, \(B =\) _____________________
    3. \(P(\text{A}) =\) _____________, \(P(\text{B}) =\) ________________
    4. \(\text{A AND B} =\) ____________________, \(\text{A OR B} =\) ________________
    5. \(P(\text{A AND B}) =\) _________, \(P(\text{A OR B}) =\) _____________
    6. \(\text{A′} =\) _____________, \(P(\text{A′}) =\) _____________
    7. \(P(\text{A}) + P(\text{A′}) =\) ____________
    8. \(P(\text{A|B}) =\) ___________, \(P(\text{B|A}) =\) _____________; are the probabilities equal?

    Answer

    1. \(\text{S} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\)
    2. \(\text{A} = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}, \text{B} = \{14, 15, 16, 17, 18, 19\}\)
    3. \(P(\text{A}) = \frac{9}{19}\), \(P(\text{B}) = \frac{6}{19}\)
    4. \(\text{A AND B} = \{14,16,18\}\), \(\text{A OR B} = \{2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19\}\)
    5. \(P(\text{A AND B}) = \frac{3}{19}\), \(P(\text{A OR B}) = \frac{12}{19}\)
    6. \(\text{A′} = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19\); \(P(\text{A′}) = \frac{10}{19}\)
    7. \(P(\text{A}) + P(\text{A′}) = 1\left((\frac{9}{19} + \frac{10}{19} = 1\right)\)
    8. \(P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}} = \frac{3}{6}, P(\text{B|A}) = \frac{\text{P(A AND B)}}{\text{P(A)}} = \frac{3}{9}\), No

    Example \(\PageIndex{2A}\)

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

    1. Event \(\text{T} =\) the outcome is two.
    2. Event \(\text{A} =\) the outcome is an even number.
    3. Event \(\text{B} =\) the outcome is less than four.
    4. The complement of \(\text{A}\).
    5. \(\text{A GIVEN B}\)
    6. \(\text{B GIVEN A}\)
    7. \(\text{A AND B}\)
    8. \(\text{A OR B}\)
    9. \(\text{A OR B′}\)
    10. Event \(\text{N} =\) the outcome is a prime number.
    11. Event \(\text{I} =\) the outcome is seven.

    Solution

    1. \(\text{T} = \{2\}\), \(P(\text{T}) = \frac{1}{6}\)
    2. \(A = \{2, 4, 6\}\), \(P(\text{A}) = \frac{1}{2}\)
    3. \(\text{B} = \{1, 2, 3\}\), \(P(\text{B}) = \frac{1}{2}\)
    4. \(\text{A′} = \{1, 3, 5\}, P(\text{A′}) = \frac{1}{2}\)
    5. \(\text{A|B} = \{2\}\), \(P(\text{A|B}) = \frac{1}{3}\)
    6. \(\text{B|A} = \{2\}\), \(P(\text{B|A}) = \frac{1}{3}\)
    7. \(\text{A AND B} = {2}, P(\text{A AND B}) = \frac{1}{6}\)
    8. \(\text{A OR B} = \{1, 2, 3, 4, 6\}\), \(P(\text{A OR B}) = \frac{5}{6}\)
    9. \(\text{A OR B′} = \{2, 4, 5, 6\}\), \(P(\text{A OR B′}) = \frac{2}{3}\)
    10. \(\text{N} = \{2, 3, 5\}\), \(P(\text{N}) = \frac{1}{2}\)
    11. A six-sided die does not have seven dots. \(P(7) = 0\).

    Example \(\PageIndex{2B}\)

    Table describes the distribution of a random sample \(S\) of 100 individuals, organized by gender and whether they are right- or left-handed.

      Right-handed Left-handed
    Males 43 9
    Females 44 4

    Let’s denote the events \(M =\) the subject is male, \(F =\) the subject is female, \(R =\) the subject is right-handed, \(L =\) the subject is left-handed. Compute the following probabilities:

    1. \(P(\text{M})\)
    2. \(P(\text{F})\)
    3. \(P(\text{R})\)
    4. \(P(\text{L})\)
    5. \(P(\text{M AND R})\)
    6. \(P(\text{F AND L})\)
    7. \(P(\text{M OR F})\)
    8. \(P(\text{M OR R})\)
    9. \(P(\text{F OR L})\)
    10. \(P(\text{M'})\)
    11. \(P(\text{R|M})\)
    12. \(P(\text{F|L})\)
    13. \(P(\text{L|F})\)

    Answer

    1. \(P(\text{M}) = 0.52\)
    2. \(P(\text{F}) = 0.48\)
    3. \(P(\text{R}) = 0.87\)
    4. \(P(\text{L}) = 0.13\)
    5. \(P(\text{M AND R}) = 0.43\)
    6. \(P(\text{F AND L}) = 0.04\)
    7. \(P(\text{M OR F}) = 1\)
    8. \(P(\text{M OR R}) = 0.96\)
    9. \(P(\text{F OR L}) = 0.57\)
    10. \(P(\text{M'}) = 0.48\)
    11. \(P(\text{R|M}) = 0.8269\) (rounded to four decimal places)
    12. \(P(\text{F|L}) = 0.3077\) (rounded to four decimal places)
    13. \(P(\text{L|F}) = 0.0833\)

    WeBWorK Problems

    Query \(\PageIndex{1}\)

    Query \(\PageIndex{2}\)

    Query \(\PageIndex{3}\)

    Query \(\PageIndex{4}\)

    Query \(\PageIndex{5}\)

    Query \(\PageIndex{6}\)

    Query \(\PageIndex{7}\)

    References

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

    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

    \(\text{A}\) and \(\text{B}\) are events

    \(P(\text{S}) = 1\) where \(\text{S}\) is the sample space

    \(0 \leq P(\text{A}) \leq 1\)

    \(P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}}\)

    Glossary

    Conditional Probability
    the likelihood that an event will occur given that another event has already occurred
    Equally Likely
    Each outcome of an experiment has the same probability.
    Event
    a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by \(S\). An event is an arbitrary subset in \(S\). It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as \(A, B, C\), and so on.
    Experiment
    a planned activity carried out under controlled conditions
    Outcome
    a particular result of an experiment
    Probability
    a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following 3 axioms (by A.N. Kolmogorov, 1930’s): Let \(S\) denote the sample space and \(A\) and \(B\) are two events in S. Then:
    • \(0 \leq P(\text{A}) \leq 1\)
    • If \(\text{A}\) and \(\text{B}\) are any two mutually exclusive events, then \(\text{P}(\text{A OR B}) = P(\text{A}) + P(\text{B})\).
    • \(P(\text{S}) = 1\)
    Sample Space
    the set of all possible outcomes of an experiment
    The AND Event
    An outcome is in the event \(\text{A AND B}\) if the outcome is in both \(\text{A AND B}\) at the same time.
    The Complement Event
    The complement of event \(\text{A}\) consists of all outcomes that are NOT in \(\text{A}\).
    The Conditional Probability of A GIVEN B
    \(P(\text{A|B})\) is the probability that event \(\text{A}\) will occur given that the event \(\text{B}\) has already occurred.
    The Or Event
    An outcome is in the event \(\text{A OR B}\) if the outcome is in \(\text{A}\) or is in \(\text{B}\) or is in both \(\text{A}\) and \(\text{B}\).

    Contributors and Attributions

    • Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.

    This page titled 4.1: Probability Experiments and Sample Spaces is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.