7.8 The Addition Rule for Probability
3 min read•june 18, 2024
The for helps us calculate the chances of multiple events happening. It's like figuring out how likely you are to win a game by adding up your different ways to win.
For events that can't happen at the same time, we just add their individual chances. But for events that could overlap, we need to subtract the chance of them happening together. This helps us avoid counting the same twice.
The Addition Rule for Probability
Addition rule for mutually exclusive events
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- States the probability of either event A or event B occurring is the sum of their individual probabilities:
- Only applies when events A and B are mutually exclusive meaning they cannot occur simultaneously (flipping a coin and getting heads or tails)
- To calculate the probability of a compound event using this rule:
- Determine the individual probabilities of each event
- Add the individual probabilities together to find the probability of the compound event
- Examples:
- If the probability of rolling a 1 on a fair die is and the probability of rolling a 2 is also , the probability of rolling either a 1 or a 2 is
- The probability of drawing a red card or a black card from a standard deck is since there are 26 red cards and 26 black cards in a 52-card deck
- The of an event A is the probability of A not occurring, denoted as P(not A) or P(A')
Inclusion/exclusion principle for probabilities
- Used to find the probability of the of two or more events that are not mutually exclusive
- For two events A and B, the principle states:
- represents the probability of both events A and B occurring simultaneously, also known as the of the events
- To apply this principle:
- Calculate the individual probabilities of each event
- Calculate the probability of the intersection of the events
- Add the individual probabilities and subtract the probability of the intersection
- Examples:
- If the probability of drawing a heart from a standard deck of cards is and the probability of drawing a red card is , the probability of drawing a heart or a red card is
- The probability of selecting a student who plays soccer or basketball from a group of 100 students, where 20 play soccer, 15 play basketball, and 5 play both sports is
- Venn diagrams can be used to visualize the relationships between sets and their intersections in probability problems
Identification of mutually exclusive events
- Events are mutually exclusive if they cannot occur at the same time meaning the intersection of the events is empty
- To determine if events are mutually exclusive:
- List the possible outcomes for each event
- Compare the outcomes to see if there are any common elements
- If there are no common elements, the events are mutually exclusive
- If there are common elements, the events are not mutually exclusive
- Examples:
- When rolling a fair die, the events "rolling an even number" and "rolling a number less than 3" are not mutually exclusive because they share the common outcome of rolling a 2
- When drawing a card from a standard deck, the events "drawing a heart" and "drawing a spade" are mutually exclusive because there are no cards that are both a heart and a spade
- Flipping a coin and getting heads or tails are since a coin cannot land on both heads and tails in a single flip
Set Theory and Probability
- provides a foundation for understanding probability concepts
- are events where the occurrence of one does not affect the probability of the other
- is the probability of an event occurring given that another event has already occurred
Key Terms to Review (23)
∩: The symbol '∩' represents the intersection of two sets, which includes all elements that are common to both sets. Understanding this concept is essential for analyzing relationships between different groups and their shared characteristics. The intersection helps in identifying overlaps and can be used to simplify problems in various mathematical contexts, including probability calculations.
∪: The symbol ∪ represents the union of two sets, which combines all the unique elements from both sets without duplication. This operation is essential in understanding how different sets can interact and form new sets, highlighting relationships between different groups of items or probabilities. The concept of union is foundational for various mathematical operations and theories, particularly in set theory and probability.
Addition Rule: The addition rule is a fundamental principle in probability that determines the likelihood of the occurrence of at least one of several events. It connects various outcomes and probabilities, particularly when events are mutually exclusive or not, and plays a key role in analyzing situations using tree diagrams and tables. Understanding the addition rule allows for effective calculation of probabilities in more complex scenarios involving permutations, combinations, and conditional probabilities.
Cardinality of the union of two sets: The cardinality of the union of two sets is the number of unique elements present in both sets combined. It is calculated by adding the cardinalities of each set and subtracting the cardinality of their intersection.
Complement: The complement of a set A, denoted by A', consists of all elements not in A but within the universal set U. The universal set U contains all possible elements under consideration.
Complement: In set theory, the complement of a set refers to all the elements in the universal set that are not included in that specific set. Understanding the complement is crucial as it helps in visualizing and analyzing relationships between sets, especially when using diagrams, performing operations with two or three sets, applying De Morgan's laws, and calculating probabilities.
Conditional Probability: Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. This concept is essential for understanding how different events can influence one another, especially when using tools like tree diagrams, tables, and outcomes to visualize probabilities, as well as when dealing with permutations and combinations.
Empirical probability: Empirical probability is the probability of an event determined by conducting experiments or observing real-life occurrences. It is calculated as the ratio of the number of favorable outcomes to the total number of trials.
Inclusion/Exclusion Principle: The Inclusion/Exclusion Principle is a fundamental concept in combinatorics and probability that allows for the calculation of the size of the union of multiple sets. This principle accounts for overlapping elements among the sets, ensuring that each element is counted only once when determining probabilities or counting outcomes. By systematically including and excluding the intersections of sets, one can accurately compute the total number of elements in unions and avoid overcounting.
Independent events: Independent events are outcomes in probability that do not influence each other; the occurrence of one event does not change the probability of the other event occurring. Understanding independent events is crucial for calculating probabilities accurately, especially when using methods like permutations and combinations, assessing odds, applying addition and multiplication rules, evaluating conditional probabilities, and computing expected values.
Intersection: The intersection of two or more sets is the set containing all elements that are common to each of the sets. This concept is crucial for understanding relationships between different groups, helping visualize shared traits or properties through various methods.
Intersection of two sets: The intersection of two sets is a new set containing all the elements that are common to both original sets. The symbol for intersection is ∩.
Mutually exclusive events: Mutually exclusive events are outcomes that cannot occur at the same time. If one event happens, it excludes the possibility of the other occurring simultaneously. This concept is fundamental in probability and helps in analyzing outcomes using various methods, making it easier to calculate the likelihood of different events happening.
Outcome: An outcome is a possible result of a random experiment or event, which can be described in terms of the various scenarios that could occur. It connects to counting techniques, probability rules, and methods for organizing and visualizing data, all of which are essential for understanding how outcomes influence decision-making and predictions in uncertain situations.
P(A and B): P(A and B) represents the probability that both events A and B occur simultaneously. This concept is crucial in understanding how different events can interact with each other, particularly when considering outcomes that depend on multiple conditions or scenarios. Recognizing how to calculate this joint probability is vital for correctly applying rules related to combinations, understanding mutual exclusivity, and working through situations where one event influences another.
P(A or B): P(A or B) represents the probability of either event A occurring, event B occurring, or both events happening simultaneously. This term is crucial when determining the likelihood of at least one of two events taking place, highlighting their relationship and the overlap in cases where both may occur. Understanding P(A or B) is essential for calculating probabilities in scenarios involving combinations of events and applying the addition rule effectively.
Probability: Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It connects various mathematical concepts by providing a framework to assess and quantify uncertainty in different scenarios, helping to determine outcomes based on different arrangements, selections, and occurrences.
Sample space: Sample space is the set of all possible outcomes in a probability experiment. It provides a comprehensive list of everything that could happen during the experiment.
Sample Space: A sample space is the set of all possible outcomes of a random experiment. Understanding the sample space is crucial because it forms the foundation for calculating probabilities, counting outcomes, and analyzing events in various contexts.
Set theory: Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It provides a foundational framework for various mathematical concepts and operations, including relationships between different groups, classifications, and how elements interact within those groups. This framework is crucial for understanding concepts like subsets, Venn diagrams, and various set operations, which are fundamental in both theoretical and applied mathematics.
Union: In set theory, the union refers to the operation that combines all distinct elements from two or more sets, creating a new set that contains every element present in any of the sets involved. This operation highlights how different collections of items can be merged together without duplication, showcasing the overall diversity of elements.
Venn diagram: A Venn diagram is a visual representation of sets and their relationships, using overlapping circles to illustrate how different sets intersect, are separate, or share common elements. This tool helps in understanding basic set concepts and is widely used in various mathematical operations involving two or more sets, including logical arguments, probabilities, and outcomes.
Venn diagram with three intersecting sets: A Venn diagram with three intersecting sets is a diagram that uses three overlapping circles to represent all possible logical relations between the sets. Each region within the diagram corresponds to different combinations of inclusion and exclusion among the sets.