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7.5 Basic Concepts of Probability

2 min read•june 18, 2024

is all about predicting outcomes. It's like having a crystal ball, but instead of magic, we use math. We'll learn how to calculate chances using ratios, data, and even gut feelings.

We'll also discover the , which helps us find probabilities by looking at what's left over. Plus, we'll touch on some fancier concepts like and . It's like becoming a mathematical fortune-teller!

Probability Fundamentals

Ratio for theoretical probabilities

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  • Assumes equally likely outcomes for events
  • Identifies total number of possible outcomes in the
  • Determines number of favorable outcomes for the specific event
  • Calculates using the ratio [P(event)](https://www.fiveableKeyTerm:P(event))=number of favorable outcomestotal number of possible outcomes[P(event)](https://www.fiveableKeyTerm:P(event)) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ possible\ outcomes}
  • Simplifies the ratio to lowest terms or converts to decimal or percentage (coin flip, P(heads)=12P(heads) = \frac{1}{2} or 0.5 or 50%)

Methods of probability assignment

  • uses mathematical reasoning assuming equally likely outcomes
    • Calculates using the ratio P(event)=number of favorable outcomestotal number of possible outcomesP(event) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ possible\ outcomes} (rolling a 3 on a fair die, P(3)=16P(3) = \frac{1}{6})
  • Empirical (experimental) probability based on data from actual experiments or simulations
    • Calculates using the ratio P(event)=number of times the event occurstotal number of trialsP(event) = \frac{number\ of\ times\ the\ event\ occurs}{total\ number\ of\ trials} (flipping a coin 100 times with 55 heads, P(heads)=55100=0.55P(heads) = \frac{55}{100} = 0.55)
    • Approaches theoretical probability as number of trials increases ()
  • relies on personal beliefs, opinions, judgments about likelihood
    • Not based on mathematical reasoning or observed data (estimating the probability of a favorite sports team winning a championship)

Complement rule in probability

  • The complement of event A (A' or Aˉ\bar{A}) is all outcomes in not in A
  • Complement rule: [P(A)](https://www.fiveableKeyTerm:P(A))=1−[P(A′)](https://www.fiveableKeyTerm:P(A′))[P(A)](https://www.fiveableKeyTerm:P(A)) = 1 - [P(A')](https://www.fiveableKeyTerm:P(A')) or P(A′)=1−P(A)P(A') = 1 - P(A)
  • Calculates probability of an event by finding probability of its complement
  • Useful when the complement is easier to calculate or more intuitive
  • Examples:
    1. Drawing an ace from a standard deck of 52 cards
      • P(ace)=452=113P(ace) = \frac{4}{52} = \frac{1}{13}, P(not ace)=1−113=1213P(not\ ace) = 1 - \frac{1}{13} = \frac{12}{13}
    2. Selecting a defective product from a batch of 1000 with 10 defects
      • P(defect)=101000=0.01P(defect) = \frac{10}{1000} = 0.01, P(not defect)=1−0.01=0.99P(not\ defect) = 1 - 0.01 = 0.99

Advanced Probability Concepts

  • Independence: Events A and B are independent if the occurrence of one does not affect the probability of the other
  • : The probability of an event A occurring, given that another event B has already occurred
  • Expected value: The average outcome of an experiment if it is repeated many times
  • : Two events that cannot occur simultaneously (e.g., rolling an even number and an odd number on a single die roll)
  • : A variable whose possible values are outcomes of a random phenomenon

Key Terms to Review (19)

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.
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.
Empirical Probability: Empirical probability is the likelihood of an event occurring based on observed data rather than theoretical calculations. This type of probability is calculated by taking the number of times an event occurs and dividing it by the total number of trials or observations. It emphasizes actual results from experiments or real-life situations, making it a practical approach to understanding probabilities in various contexts.
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.
Experimental Probability: Experimental probability is the likelihood of an event occurring based on actual experiments or trials, as opposed to theoretical calculations. This approach uses real-world data to calculate the probability of an event, making it a practical way to understand outcomes. It highlights the difference between expected results and observed results, which can vary significantly due to random chance.
Independence: Independence refers to the scenario where the occurrence of one event does not affect the probability of another event occurring. This concept is crucial as it underlies many basic principles in probability, influencing how we calculate probabilities of combined events and affecting distributions such as the binomial distribution.
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.
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.
P(A'): P(A') represents the probability of the complement of event A occurring. In probability theory, this is crucial because it helps in understanding outcomes that are not included in event A, allowing for a comprehensive analysis of all possible outcomes in a given scenario. The relationship between an event and its complement is foundational in probability, as it enables calculations of probabilities for multiple events and their interactions.
P(A): P(A) represents the probability of event A occurring. This concept is foundational in probability theory, linking closely to the likelihood of specific outcomes and their relation to the total possible outcomes in a sample space. Understanding P(A) is crucial for calculating probabilities in various contexts, including independent and dependent events.
P(event): P(event) refers to the probability of a specific event occurring, expressed as a number between 0 and 1. This concept helps quantify uncertainty, allowing for the analysis of various outcomes in a random experiment. Understanding P(event) is crucial for making predictions and informed decisions based on likelihoods.
Probability: Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means it will definitely happen.
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.
Random variable: A random variable is a numerical outcome of a random process, serving as a way to quantify uncertainty. It can take on different values based on the outcomes of a particular experiment or event, and is often used to model real-world scenarios. Random variables can be classified into two types: discrete and continuous, and they are foundational to understanding probability distributions and statistical measures.
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.
Subjective probability: Subjective probability is the measure of an individual's personal judgment or belief about the likelihood of a particular event occurring. Unlike objective probabilities, which rely on empirical data and statistical methods, subjective probabilities are influenced by personal experiences, opinions, and information that may not be quantifiable. This type of probability is often used in situations where there is uncertainty and a lack of historical data.
Theoretical probability: Theoretical probability is the calculated likelihood of an event occurring based on the possible outcomes in a given situation. This type of probability assumes that all outcomes are equally likely and is often expressed as a fraction, decimal, or percentage. It forms the basis for understanding how likely events are in various scenarios, helping to make predictions and informed decisions.
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7.6 Probability with Permutations and Combinations