Glossary

7.6 Probability with Permutations and Combinations

3 min read•june 18, 2024

concepts help us calculate the likelihood of events occurring. We explore for ordered scenarios and for unordered ones, using formulas to determine possible outcomes and their probabilities.

Applying probability rules allows us to solve complex problems. We use multiplication and addition rules for different event types, and learn to break down complex scenarios into manageable steps, combining results to find final probabilities.

Probability Concepts

Permutations for ordered probabilities

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  • Permutations calculate probabilities when order matters (drawing letters from Scrabble tiles to form a word)
  • Formula for permutations: [P(n,r)](https://www.fiveableKeyTerm:P(n,r))=[n!](https://www.fiveableKeyTerm:n!)(n−r)![P(n,r)](https://www.fiveableKeyTerm:P(n,r)) = \frac{[n!](https://www.fiveableKeyTerm:n!)}{(n-r)!}
    • nn represents total number of items to choose from
    • rr represents number of items being chosen
  • Calculating probability of a specific :
    • Determine number of permutations satisfying desired outcome
    • Divide this number by total possible permutations
  • Examples:
    • Arranging books on a shelf
    • Assigning tasks to team members
    • Determining possible PIN codes

Combinations in unordered scenarios

  • Combinations calculate probabilities when order is irrelevant (drawing a hand of cards from a standard deck)
  • Formula for combinations: [C(n,r)](https://www.fiveableKeyTerm:C(n,r))=n!r!(n−r)![C(n,r)](https://www.fiveableKeyTerm:C(n,r)) = \frac{n!}{r!(n-r)!}
    • nn represents total number of items to choose from
    • rr represents number of items being chosen
  • Calculating probability of a specific :
    • Determine number of combinations satisfying desired outcome
    • Divide this number by total possible combinations
  • Examples:
    • Selecting a committee from a group of people
    • Choosing toppings for a pizza
    • Forming teams for a tournament

Applying Probability Rules

Multiplication rule for complex probabilities

  • for : P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
    • Probability of both events occurring equals product of individual probabilities
  • Multiplication rule for : 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))
    • Probability of both events occurring equals product of first event's probability and of second event given first occurred
  • for : P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
    • Probability of either event occurring equals sum of individual probabilities
  • Addition rule for : 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)
    • Probability of either event occurring equals sum of individual probabilities minus probability of both events occurring
  • : P(A′)=1−P(A)P(A') = 1 - P(A)
    • Probability of an event not occurring equals 1 minus the probability of it occurring
  • Solving complex probability problems:
    1. Break problem into smaller, manageable steps
    2. Identify whether permutations or combinations are required for each step
    3. Apply appropriate probability rules (multiplication, addition) as needed
    4. Combine results of each step to determine final probability
  • Examples:
    • Calculating probability of drawing specific cards from a deck
    • Determining likelihood of multiple events occurring in a sequence
    • Analyzing probability of winning a lottery

Additional Probability Concepts

  • : The set of all possible outcomes in a probability experiment
  • Conditional probability: The probability of an event occurring given that another event has already occurred
  • : The average outcome of an experiment if it is repeated many times
  • : As the number of trials in an experiment increases, the experimental probability approaches the theoretical probability

Key Terms to Review (25)

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.
C(n,r): C(n,r) represents the number of ways to choose 'r' items from a total of 'n' items without regard to the order of selection. This is known as a combination and is crucial for calculating probabilities in situations where the arrangement of selected items does not matter. Understanding C(n,r) allows for deeper insights into various probability problems, especially when distinguishing between combinations and permutations.
Combination: A combination is a selection of items from a larger set where the order of selection does not matter. Understanding combinations helps in various scenarios such as calculating probabilities, forming groups, and organizing outcomes where the sequence is irrelevant, linking directly to concepts like counting rules, permutations, and probability calculations.
Combinations: Combinations refer to the selection of items from a larger set where order does not matter. They are used to determine how many ways a subset of items can be chosen from the entire set without regard to the sequence of selection.
Complement rule: The complement rule in probability refers to the principle that the probability of an event not occurring is equal to one minus the probability of the event occurring. This concept allows for easier calculation of probabilities by focusing on what does not happen rather than what does. Understanding this rule is essential when dealing with complex probability scenarios, as it often simplifies calculations and helps in interpreting results.
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.
Dependent events: Dependent events are occurrences where the outcome of one event affects the outcome of another event. This connection means that the probability of one event happening is influenced by whether or not the other event has already occurred. Understanding dependent events is crucial when calculating probabilities, especially when using combinations and permutations or applying conditional probabilities and the multiplication rule.
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.
Expected value: Expected value is a fundamental concept in probability that represents the average outcome of a random event over a large number of trials. It is calculated by multiplying each possible outcome by its probability and summing the results.
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.
Law of large numbers: The law of large numbers is a statistical theorem that states that as the number of trials or observations increases, the sample mean will get closer to the expected value or population mean. This principle highlights the reliability of averages and ensures that larger samples provide a more accurate representation of the overall population, making it essential in probability and statistics.
Multiplication Rule: The Multiplication Rule is a fundamental principle in counting and probability that states if there are multiple independent events, the total number of possible outcomes can be found by multiplying the number of choices for each event. This rule connects different aspects of combinatorial counting, outcome analysis, and probability calculations, allowing us to determine the likelihood of various outcomes occurring together.
Multiplication Rule for Counting: The Multiplication Rule for Counting is a fundamental principle used to determine the total number of possible outcomes in a sequence of events. It states that if one event can occur in \(m\) ways and a second event can occur independently in \(n\) ways, then the two events together can occur in \(m imes n\) ways.
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.
N!: The notation 'n!' represents the factorial of a non-negative integer n, defined as the product of all positive integers less than or equal to n. This concept is crucial in combinatorial mathematics, as it is used to calculate the number of ways to arrange or select items from a set, forming the foundation for understanding arrangements and selections in various contexts.
Non-mutually exclusive events: Non-mutually exclusive events are events that can occur simultaneously, meaning that the occurrence of one event does not preclude the occurrence of another. In probability, this concept is essential when calculating the likelihood of various outcomes, particularly when using permutations and combinations, as it allows for overlapping possibilities in the sample space.
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.
P(B|A): P(B|A) represents the conditional probability of event B occurring given that event A has already occurred. This concept is crucial in understanding how probabilities change based on prior events, especially when dealing with permutations and combinations where the arrangement or selection of items can affect the likelihood of outcomes. It allows for a more precise calculation of probabilities in situations where certain conditions or restrictions are in place.
P(n,r): P(n,r) represents the number of ways to arrange 'r' objects selected from a total of 'n' distinct objects, where the order of selection matters. This concept is crucial in understanding permutations, as it helps calculate how many different sequences can be formed when choosing and arranging items from a larger set. P(n,r) highlights the importance of order in arrangements, differentiating it from combinations where order is not considered.
Permutation: A permutation is an arrangement of objects in a specific order. The concept of permutations is essential for understanding how to count and organize different sequences, especially when considering distinct groups of items. Permutations are closely related to the multiplication rule, as the number of ways to arrange objects can often be calculated by multiplying the number of choices available at each step.
Permutations: Permutations are arrangements of objects in a specific order. The order of the objects is crucial and changing the order creates a different permutation.
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.
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