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3.2 Independent and Mutually Exclusive Events

3 min readLast Updated on June 27, 2024

Probability events can be independent or mutually exclusive, but rarely both. Independent events don't affect each other's likelihood, while mutually exclusive events can't happen together. Understanding these concepts is crucial for calculating probabilities accurately.

Sampling with and without replacement impacts event independence. With replacement keeps probabilities constant, while without replacement changes them. These ideas form the foundation for more advanced probability concepts like the Law of Total Probability and Bayes' Theorem.

Independent and Mutually Exclusive Events

Independence vs mutual exclusivity

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  • Independent events occur when the outcome of one event does not influence the probability of another event happening (rolling a 6 on a die, then flipping a coin and getting heads)
    • Probability of both events occurring together is the product of their individual probabilities: P(AB)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
    • Conditional probability of one event given the other remains unchanged: [P(AB)](https://www.fiveableKeyTerm:P(AB))=P(A)[P(A|B)](https://www.fiveableKeyTerm:P(A|B)) = P(A) and P(BA)=P(B)P(B|A) = P(B)
  • Mutually exclusive events cannot happen at the same time (drawing a heart or a spade from a deck of cards in a single draw)
    • Probability of both events occurring simultaneously is always zero: P(AB)=0P(A \cap B) = 0
    • Probability of either event occurring is the sum of their individual probabilities: P(AB)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
  • Events can only be both independent and mutually exclusive if at least one event has a probability of zero (rolling a 7 on a standard six-sided die, then flipping a coin and getting tails)
  • Venn diagrams can be used to visually represent the relationship between independent and mutually exclusive events

Sampling with and without replacement

  • Sampling with replacement involves putting each selected item back before the next selection, keeping event probabilities constant (picking a marble from a bag, recording its color, then putting it back and repeating)
    • Each selection is an independent event
    • Probability of an event remains the same for each pick
  • Sampling without replacement involves keeping selected items out for subsequent selections, changing event probabilities (dealing cards from a shuffled deck)
    • Probability of an event changes after each selection
    • Use the hypergeometric probability formula to calculate probabilities:
      • [P(X = k)](https://www.fiveableKeyTerm:P(X_=_k)) = \frac{{K \choose k} {N-K \choose n-k}}{{N \choose n}}
      • NN = total items in the population (52 cards in a standard deck)
      • KK = items with desired characteristic in the population (13 hearts in a standard deck)
      • nn = sample size (5 card poker hand)
      • kk = items with desired characteristic in the sample (2 hearts in a 5 card hand)

Classification of probability events

  • Determine event independence by checking if:
    1. P(AB)=P(A)P(A|B) = P(A) and P(BA)=P(B)P(B|A) = P(B)
    2. P(AB)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
    • If both conditions hold true, the events are independent (drawing a king, then drawing a heart from a standard deck with replacement)
  • Determine event mutual exclusivity by checking if:
    1. The events cannot occur at the same time
    2. P(AB)=0P(A \cap B) = 0
    • If both conditions hold true, the events are mutually exclusive (drawing a king or drawing a queen from a standard deck in a single draw)
  • Verify event independence and mutual exclusivity based on given probability values and event descriptions to accurately classify the relationship between events
  • Independence tests can be used to statistically verify the independence of events in more complex scenarios

Advanced Probability Concepts

  • Law of Total Probability: Used to calculate the probability of an event by considering all possible scenarios
  • Bayes' Theorem: Allows for updating probabilities based on new information, connecting prior and posterior probabilities
  • These concepts build upon the understanding of independent and mutually exclusive events to solve more complex probability problems

Key Terms to Review (23)

Addition Rule: The addition rule is a fundamental concept in probability theory that allows for the calculation of the probability of the occurrence of one or more mutually exclusive events. It provides a way to determine the probability of the union of two or more events when they are independent or mutually exclusive.
Bayes' Theorem: Bayes' Theorem is a fundamental concept in probability and statistics that describes the likelihood of an event occurring given the prior knowledge of the conditions related to that event. It provides a way to update the probability of a hypothesis as more information or evidence becomes available.
Binomial Distribution: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. It is a fundamental concept in probability theory and statistics, with applications across various fields.
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 or occurrence of another related event.
Contingency Tables: A contingency table, also known as a cross-tabulation or two-way table, is a statistical tool used to display and analyze the relationship between two or more categorical variables. It provides a way to investigate the association or dependence between these variables by organizing the data into a tabular format.
Hypergeometric Probability: Hypergeometric probability is a discrete probability distribution that describes the probability of a certain number of successes in a fixed number of trials, without replacement, from a finite population. It is particularly useful in situations where the population size is relatively small, and the probability of success in each trial is not constant across trials.
Independence Tests: Independence tests are statistical methods used to determine whether two or more events or variables are independent of each other. They are essential in analyzing the relationships between different phenomena and making informed decisions based on data.
Independent Events: Independent events are events whose outcomes do not influence or depend on the outcomes of other events. The occurrence of one event does not affect the probability of the other event occurring.
Joint Probability: Joint probability is the probability of two or more events occurring together or simultaneously. It represents the likelihood of multiple events happening concurrently within a given scenario or experiment.
Law of Total Probability: The law of total probability is a fundamental concept in probability theory that describes how the probability of an event can be calculated by considering the probabilities of mutually exclusive and exhaustive events. It provides a framework for understanding and calculating the probability of an event when the sample space can be divided into distinct, non-overlapping subsets.
Multiplication Rule: The multiplication rule, also known as the product rule, is a fundamental concept in probability theory that describes the probability of the intersection of two or more independent events. It provides a way to calculate the probability of multiple events occurring together by multiplying their individual probabilities.
Mutually Exclusive Events: Mutually exclusive events are events that cannot occur simultaneously or together. If one event happens, the other event(s) cannot happen at the same time. This concept is central to understanding probability and how to calculate the likelihood of events occurring.
P(A ∩ B): P(A ∩ B) represents the probability of the intersection of two events, A and B. It is the probability that both event A and event B occur simultaneously. This term is crucial in understanding the concepts of independent and mutually exclusive events.
P(A ∪ B): P(A ∪ B) represents the probability of the union of events A and B. The union of two events refers to the occurrence of either event A, event B, or both events A and B. It is the sum of the individual probabilities of the two events, minus the probability of their intersection.
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 that both events A and B will happen together, and is a fundamental concept in the study of probability and statistics.
P(A or B): P(A or B) is the probability of the occurrence of either event A or event B, or both. It represents the likelihood that at least one of the two events will occur. This concept is fundamental in understanding the basic rules of probability and how to calculate the probability of combined events.
P(A|B): P(A|B), or the conditional probability of A given B, is the probability of event A occurring, given that event B has already occurred. It represents the likelihood of one event happening, given the occurrence of another event. This concept is central to understanding the relationships between events and their probabilities, particularly in the context of independent and mutually exclusive events.
P(X = k): P(X = k) represents the probability that a random variable X takes on a specific value k. It is a fundamental concept in probability theory and statistics, used to quantify the likelihood of a particular outcome occurring in a given scenario.
Probability: Probability is the measure of the likelihood that an event will occur. It quantifies the chance or possibility of a particular outcome happening within a given set of circumstances or a defined sample space. Probability is a fundamental concept in statistics, as it provides a framework for understanding and analyzing uncertain events and their associated likelihoods.
Sample Space: The sample space refers to the set of all possible outcomes or results in a probability experiment. It represents the universal set of all possible events or scenarios that can occur in a given situation. The sample space is a fundamental concept in probability theory that provides the foundation for understanding and calculating probabilities.
Sampling with Replacement: Sampling with replacement is a statistical technique where items are selected from a population, and after each selection, the item is returned to the population before the next selection is made. This means that the same item can be selected multiple times during the sampling process, allowing for the possibility of repetition in the sample.
Sampling Without Replacement: Sampling without replacement is a method of drawing samples from a population where each item is removed from the population after it is selected, ensuring that no item can be selected more than once. This is in contrast to sampling with replacement, where items are returned to the population after being selected, allowing for the possibility of the same item being chosen multiple times.
Venn Diagrams: Venn diagrams are visual representations that use overlapping circles to illustrate the relationships between different sets or groups. They are commonly used to analyze probabilities, explore logical relationships, and compare and contrast concepts.
Addition Rule
See definition

The addition rule is a fundamental concept in probability theory that allows for the calculation of the probability of the occurrence of one or more mutually exclusive events. It provides a way to determine the probability of the union of two or more events when they are independent or mutually exclusive.

Term 1 of 23

How does sampling without replacement affect the probability of selecting a specific item in subsequent draws compared to sampling with replacement?

1 of 2
Addition Rule
See definition

The addition rule is a fundamental concept in probability theory that allows for the calculation of the probability of the occurrence of one or more mutually exclusive events. It provides a way to determine the probability of the union of two or more events when they are independent or mutually exclusive.

Term 1 of 23

How does sampling without replacement affect the probability of selecting a specific item in subsequent draws compared to sampling with replacement?

1 of 2


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.