Key Concepts Study Tool: Chapter 04

Key Concepts Study Tool: Chapter 04

Click on each concept below to check your understanding.

1. Probability

  • A value between 0 and 1 (or 0 and 100 per cent, when stated as a percentage)
  • Where: 0 = an event that never occurs; 1 = an event that definitely occurs
  • probability of an event = 1 – (the probability that it will not occur)

2. Probability: Sample Space, Random Variables, Trial, Experiment, Law of Large Numbers

  • Sample Space: Contains all of the theoretically possible outcomes of an event. Each probability is a fraction of the sample space.
  • Random Variables: The value is subject to variation from known or unknown sources and has a value that is the result of a process or experiment (e.g., tossing a coin) within an already known sample space.
  • Trial: An individual exercise that, when taken alongside other exercises, will collectively form the data for the experiment.
  • Experiment: A collection of exercises or trials.
  • Law of Large Numbers: Assumes that if a random experiment is repeated many times then the outcomes will eventually reach a level of “stability.” Therefore, the more times you perform a trial, the closer you’ll be to your calculated theoretical probability.

3. Types of Probabilities

  • Empirical probability: The estimated probability of an event calculated by using real data from an experiment.
  • Theoretical probability: The probability of an occurrence, based on logical reasoning or deduction.
  • Discrete probability: Clearly defined, non-overlapping outcomes. Variables with discrete probabilities often have an equal probability of occurrence for each value. The sum of probabilities is always equal to 1.

4. The Probability of Unrelated Events

  • Events are independent, such that the first event has no impact on the outcome of the second.
  • Multiplication rule of probabilities – observing two independent outcomes in succession is equal to the product of the probability of the two individual outcomes.
  • p(A and B) = p(A)*p(B)

5. Mutually Exclusive Probabilities That Are Interchangeable

  • Mutually exclusive events are events that cannot occur together.
  • Addition rule of probabilities: To determine the probability of either of two mutually exclusive events occurring, it’s necessary to add the independent probabilities.
  • p(A or B) = p(A)+p(B)

6. Non-Mutually Exclusive Probabilities

  • When two events can occur simultaneously, they are considered non-mutually exclusive, or interchangeable, probabilities.
  • Calculating this type of probability requires subtracting all duplications to avoid over counting.
  • p(A or B) = p(A) + p(B) – p(A and B)

7. Calculating a Discrete Theoretical Probability

  • Identify the experiment of interest (coin tossing, card drawing, etc.), and be sure that the outcomes are mutually exclusive.
  • Determine the sample set, or the total number of possible outcomes. This will be the denominator of your probability calculations.
  • Determine the frequency of occurrence of your outcome of interest (a coin landing on heads, drawing a spade, etc.).This will be the numerator of your calculation.
  • Divide the numerator by the denominator to determine the discrete probability.
  • Convert the probability to a percentage by multiplying the number by 100 if you wish, or leave it as a decimal.

8. Calculating Probabilities for Non-Mutually Exclusive Events

  • Calculate the probability of event A.
  • Calculate the probability of event B.
  • Subtract the number of duplications.
  • The formula takes the form p(A or B) = p(A) + p(B) – p(A and B)
  • If you prefer to see probabilities expressed as percentages, multiply the result by 100.
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