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Probability and Independence

  • Page ID
    217
  • [ "article:topic", "Venn diagram", "Conditional Probability", "Independent Events", "mutually exclusive", "counting rule" ]

    For an experiment we define an event to be any collection of possible outcomes. A simple event is an event that consists of exactly one outcome.

    • "or" means the union (i.e. either can occur)
    • "and" means intersection (i.e. both must occur)

    Two events are mutually exclusive if they cannot occur simultaneously. For a Venn diagram, we can tell that two events are mutually exclusive if their regions do not intersect

    Definition: Probability

    We define probability of an event \(E\) to be to be 

    \[P(E) = \dfrac{\text{number of simple events within E}}{\text{ total number of possible outcomes}} \]

    We have the following:

    1. \(P(E)\) is always between 0 and 1.
    2. The sum of the probabilities of all simple events must be 1.
    3. \(P(E) + P(\text{not } E) = 1\)
    4. If \(E\) and \(F\) are mutually exclusive then

    \[ P(E \text{ or } F) = P(E) + P(F) \]

    The Difference Between "and" and "or"

    If \(E\) and \(F\) are events then we use the terminology

    \[ E \text{ and } F \]

    to mean all outcomes that belong to both \(E\) and \(F\).

    We use the terminology

    \[E \text{ or } F \]

    to mean all outcomes that belong to either \(E\) or \(F\).

    Below is an example of two sets, \(A\) and \(B\), graphed in a Venn diagram.

    The green area represents \(A\) and \(B\) while all areas with color represent \(A\) or \(B\)

    Example \(\PageIndex{1}\)

    Our Women's Volleyball team is recruiting for new members. Suppose that a person inquires about the team.

    • Let \(E\) be the event that the person is female
    • Let \(F\) be the event that the person is a student

    then \(E\) and \(F\) represents the qualifications for being a member of the team. Note that \(E\) or \(F\) is not enough.

    We define

    Definition: Conditional Probability

    \[ P(E|F) = \dfrac{P(E \text{ and } F}{P(F)}\]

    We read the left hand side as "The probability of event \(E\) given event \(F\) occurred."

    We call two events independent if the following definitions hold.

    Definition: Independence

    For independent Events

    \[P(E|F) = P(E) \label{1a}\]

    Equivalently, we can say that \(E\) and \(F\) are independent if

    Definition: The Multiplication Rule

    For Independent Events

    \[P(E \text{ and } F) = P(E)P(F) \label{1b}\]

    Example \(\PageIndex{2}\)

    Consider rolling two dice. Let 

    • \(E\) be the event that the first die is a 3.
    • \(F\) be the event that the sum of the dice is an 8.

    Then \(E\) and \(F\) means that we rolled a three and then we rolled a 5

    This probability is 1/36 since there are 36 possible pairs and only one of them is (3,5)

    We have 

    \[ P(E) = 1/6 \]

    And note that (2,6),(3,5),(4,4),(5,3), and (6,2) give \(F\)

    Hence 

    \[ P(F) = 5/36\]

    We have 

    \[ P(E) P(F) = (1/6) (5/36) \]

     which is not 1/36 and we can conclude that \(E\) and \(F\) are not independent.

    Exercise \(\PageIndex{2}\)

    Test the following two events for independence:

    • \(E\) the event that the first die is a 1.
    • \(F\) the event that the sum is a 7.

    A Counting Rule

    For two events, \(E\) and \(F\), we always have

     \[ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \label{2}\]

    Example \(\PageIndex{3}\)

    Find the probability of selecting either a heart or a face card from a 52 card deck.

    Solution

    We let

    • \(E\) = the event that a heart is selected 
    • \(F\) = the event that a face card is selected

    then 

    \[ P(E) = \dfrac{1}{4} \]

    and

    \[ P(F) = \dfrac{3}{13} \]

    that is, Jack, Queen, or King out of 13 different cards of one kind.

    \[ P(E \text{ and } F) = \dfrac{3}{52} \]

    The counting rule formula (eq. 2) gives

    \[ P(E \text{ or } F) = \dfrac{1}{4} + \dfrac{3}{13} - \dfrac{3}{52} = \dfrac{22}{52} = 42\text{%}\]

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