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3.3 Two Basic Rules of Probability

3 min read•june 25, 2024

Probability rules are essential for understanding how events interact and calculating their likelihood. These rules help us determine the chances of multiple events occurring together or at least one happening among several possibilities.

The is used for calculating the probability of combined events, while the helps determine the likelihood of at least one event occurring. Understanding these rules is crucial for solving complex probability problems in various real-world scenarios.

Probability Rules

Fundamental Concepts

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  • form the foundation of probability theory
  • represents all possible outcomes of an experiment
  • An event is a subset of the
  • of an event A is all outcomes in the sample space not in A

Multiplication rule for probabilities

  • Multiplication rule applies when calculating the probability of two or more events occurring together
  • have no influence on each other's probability
    • Probability of both A and B occurring equals the product of their individual probabilities
    • Formula: P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
    • Example: Probability of rolling a 6 on a fair die (16\frac{1}{6}) and flipping a head on a fair coin (12\frac{1}{2}) equals 16×12=112\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}
  • influence each other's probability
    • Probability of both dependent events A and B occurring equals the probability of event A multiplied by the of event B given event A has occurred
    • Formula: P(A and B)=P(A)×[P(B∣A)](https://www.fiveableKeyTerm:P(B∣A))P(A \text{ and } B) = P(A) \times [P(B|A)](https://www.fiveableKeyTerm:P(B|A))
    • P(B∣A)P(B|A) represents the probability of event B occurring given that event A has already occurred
    • Example: Probability of drawing a heart from a standard deck of cards (1352\frac{13}{52}) and then drawing a king from the remaining cards (451\frac{4}{51}) equals 1352×451=151\frac{13}{52} \times \frac{4}{51} = \frac{1}{51}

Addition rule in probability

  • Addition rule applies when calculating the probability of at least one of two or more events occurring
  • cannot occur simultaneously
    • Probability of either event A or B occurring equals the sum of their individual probabilities
    • Formula: P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
    • Example: Probability of rolling an even number (12\frac{1}{2}) or a 3 (16\frac{1}{6}) on a fair die equals 12+16=23\frac{1}{2} + \frac{1}{6} = \frac{2}{3}
  • can occur simultaneously
    • Probability of either non-mutually exclusive event A or B occurring equals the sum of their individual probabilities minus the probability of both events occurring together
    • Formula: P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
    • P(A and B)P(A \text{ and } B) represents the probability of both events A and B occurring, calculated using the multiplication rule
    • Example: Probability of drawing a heart (1352\frac{13}{52}) or a face card (1252\frac{12}{52}) from a standard deck equals 1352+1252−352=1126\frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{11}{26}, as there are 3 face cards that are also hearts

Probability rules for events

  • Multiplication rule used when calculating the probability of two or more events occurring together ("and")
    • Events can be independent or dependent
    • Keywords: "and," "given," "if"
  • Addition rule used when calculating the probability of at least one of two or more events occurring ("or")
    • Events can be mutually exclusive or non-mutually exclusive
    • Keywords: "or," "either," "at least one"
  • Examples:
    1. Multiplication rule (independent events): Probability of rolling a 4 on a fair die and then flipping a tail on a fair coin
    • Conditional probability is used when events are dependent
    1. Multiplication rule (dependent events): Probability of drawing a spade from a standard deck and then drawing a queen from the remaining cards
    2. Addition rule (mutually exclusive events): Probability of rolling a prime number or a 6 on a fair die
    3. Addition rule (non-mutually exclusive events): Probability of drawing a red card or a king from a standard deck

Visualization Tools

  • help illustrate relationships between events in a sample space
  • Probability trees are useful for visualizing sequential events and their probabilities

Key Terms to Review (21)

Addition Rule: The addition rule is a fundamental concept in probability theory that describes the relationship between the probabilities of mutually exclusive events. It states that the probability of the union of two or more mutually exclusive events is equal to the sum of their individual probabilities.
Complement: The complement of an event A in probability is the set of all outcomes in the sample space that are not in A. It is often denoted as $A^c$ or $\overline{A}$.
Complement: The complement of an event is the set of all outcomes that are not part of the original event. It represents the outcomes that do not occur when the original event takes place. The concept of complement is crucial in understanding the two basic rules of probability and the interpretation of Venn diagrams.
Conditional Probability: Conditional probability is the likelihood of an event occurring given that another event has already occurred. It represents the probability of one event happening, given the knowledge of another event happening.
Conditional probability of A given B: Conditional probability of A given B, denoted as $P(A|B)$, is the probability that event A occurs given that event B has already occurred. It quantifies the relationship between two events in a probabilistic context.
Dependent Events: Dependent events are events where the outcome of one event affects the probability of the occurrence of another event. The probability of one event depends on the outcome of the other event.
Event: In the context of probability and statistics, an event is a specific outcome or set of outcomes of an experiment or random process. Events are the building blocks for understanding and analyzing probability, as they represent the possible results or occurrences that can happen in a given situation.
Independent events: Independent events are two or more events where the occurrence of one does not affect the probability of the other occurring. In mathematical terms, events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$.
Independent Events: Independent events are two or more events that have no influence on each other's outcomes. The occurrence of one event does not affect the probability of the other event occurring.
Multiplication Rule: The multiplication rule, also known as the product rule, is a fundamental concept in probability theory that describes the relationship between the probabilities of two or more independent events. It states that the probability of the joint occurrence of multiple independent events is equal to the product of their individual probabilities.
Mutually exclusive: Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot.
Mutually Exclusive Events: Mutually exclusive events are a set of events where the occurrence of one event prevents the occurrence of the other events. In other words, if one event happens, the other events cannot happen simultaneously.
Non-Mutually Exclusive Events: Non-mutually exclusive events are events that can occur simultaneously or independently of one another. This means that the occurrence of one event does not preclude the occurrence of another event. In the context of probability, non-mutually exclusive events can have overlapping outcomes, allowing for the possibility of multiple events happening at the same time.
P(A and B): P(A and B) is the probability of the intersection of two events, A and B, occurring simultaneously. It represents the likelihood of both events A and B happening together, rather than the probability of either event occurring independently.
P(A or B): P(A or B) is the probability of the occurrence of either event A or event B, or both, in a given scenario. It represents the combined likelihood of either event happening, taking into account the possibility that the events may be mutually exclusive or overlapping.
P(B|A): P(B|A) is a conditional probability that represents the probability of event B occurring, given that event A has already occurred. It is a fundamental concept in probability theory and is essential for understanding the relationship between two events.
Probability Axioms: Probability axioms are the fundamental rules that define the mathematical foundation of probability theory. These axioms provide a framework for calculating and understanding the likelihood of events occurring in a given scenario.
Probability Tree: A probability tree is a graphical representation of the possible outcomes and their associated probabilities in a probabilistic scenario. It provides a visual aid to understand the relationships between events and the likelihood of their occurrence.
Sample space: Sample space is the set of all possible outcomes of a probability experiment. It provides a comprehensive list of every potential result that can occur.
Sample Space: The sample space is the set of all possible outcomes or results of a random experiment or event. It represents the complete set of possibilities that could occur in a given situation.
Venn Diagrams: A Venn diagram is a visual representation of the relationships between different sets or groups. It uses overlapping circles to illustrate the commonalities and differences between the sets, making it a useful tool for understanding probability and set theory concepts.
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3.4 Contingency Tables and Probability Trees