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ZingPath: Probability

The Concept of Probability

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The Concept of Probability

Geometry

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Find the probability of an event happening and not happening, impossible and certain events and explain less likely and more likely events.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • After completing this Activity Object, learners will be able to:
  • Find the probability of a simple event.
  • Find the probability of a simple event.
  • Use the relationship of probabilities of an event happening and not happening.
  • Describe the likelihood of an event as less likely or more likely.
  • Describe the likelihood of an event as impossible or certain.
  • Explain that probability of an event is always between 0 and 1.

Everything You'll Have Covered

The concept of probability

A sample space is the set of all possible outcomes of an experiment. It can be denoted by the letter U.

For example, when a die is rolled, the experiment has a sample space with 6 possible outcomes and can be written as:

U = {1, 2, 3, 4, 5, 6}

An event is any subset of the sample space of an experiment and is denoted by any capital letter, such as A.

For example, when a die is rolled, it will come up 1 or 2 or 3 or 4 or 5 or 6. It is not possible to roll more than one of these numbers at the same time, and it is not possible to roll a number larger than 6.

Event A = {6} Rolling a 6

Event B = {1, 2, 3, 4, 5} Not rolling a 6

An outcome is the result of a single trial of an experiment.

For example,

  • When a coin is tossed, heads is a possible outcome.
  • When a die is rolled, 6 is a possible outcome.
  • Probability is a numerical measure of how likely it is that an event will occur. The value of probability always lies between 0 and 1.

    In notation,

  • P(E) (read P of E) stands for the probability of event E.
  • n(E) stands for the number of ways E can happen.
  • n(U) stands for the number of elements in the sample space U.
  • So the formula for probability is:

    P(E) =

    For example, a basket contains 30 apples, 20 pears and 10 peaches. What is the probability that the first piece of fruit taken from the basket will be a peach?

  • Total number of fruits in the basket = 30 + 20 + 10 = 60
  • Number of peaches in the basket = 10
  • Equally likely vs. not equally likely

    In the experiment of rolling a six-sided die, the probability of each outcome is always the same and all of the outcomes are equally likely to happen.

    P(1) = P(2) = P(3) = P(4) = P(5) = P(6) =

    Now let's examine the probability of picking each color marble from a jar containing 6 red, 5 green, 8 blue, and 3 yellow marbles.

    The possible outcomes of this experiment are red, green, blue, and yellow.

  • P(red) =
  • P(green) =
  • P(blue) =
  • P(yellow) =
  • In this experiment, the outcomes are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

    P(blue) > P(red) > P(green) > P(yellow)

    Probability of an event not happening

    To find the probability of not picking a certain color marble, we can use the formula of probability.

    For example,

    What is the probability that we would not pick a red marble from the jar?

    The sum of the probability of an event happening and the probability of it not happening is always 1:

  • P(E) + P(E') = 1
  • The probability of an event not happening is found by calculating 1 - the probability of the event happening:

  • P(E') = 1 - P(E)
  • For example,

    What is the probability of not picking a red marble from the jar?

    The above examples demonstrate finding the complement of an event.

    Certain vs. impossible

    A certain event is an event that is certain to occur. The probability of a certain event is always 1.

    For example,

    A teacher chooses a student at random from a class of 30 girls. What is the probability that the student chosen is a girl?

  • The number of students in the class is 30. So the number of possible outcomes is 30. Since all the students in the class are girls, the number of favorable outcomes is 30.
  • The probability that the student chosen is a girl is 1.

    An impossible event is an event that has no chance of occurring. The probability of an impossible event is always zero (0).

    For example,

    When a die is thrown once, the probability of getting a number greater than 6 is an impossible event, since the highest number on a die is 6.

    Tutorial Details

    Approximate Time 30 Minutes
    Pre-requisite Concepts The concept of ratio
    Course Geometry
    Type of Tutorial Concept Development
    Key Vocabulary certain event, event, formula of probability