{"id":80,"date":"2023-01-25T16:34:02","date_gmt":"2023-01-25T16:34:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/computing-the-probability-of-an-event-learn-it-page-2\/"},"modified":"2024-03-22T18:19:35","modified_gmt":"2024-03-22T18:19:35","slug":"computing-the-probability-of-an-event-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/computing-the-probability-of-an-event-learn-it-2\/","title":{"raw":"Computing the Probability of an Event: Learn It 2","rendered":"Computing the Probability of an Event: Learn It 2"},"content":{"raw":"

Basic Probability<\/h2>\r\n

The probability of an event is determined by dividing the count of favorable outcomes by the count of all possible outcomes, assuming each outcome has an equal chance of occurring.<\/p>\r\n

\r\n
\r\n

basic probability<\/h3>\r\n

Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[\/latex] using this formula:<\/p>\r\n

 <\/p>\r\n

[latex]P(E)=\\frac{\\text{Number of outcomes corresponding to the event E}}{\\text{Total number of equally-likely outcomes}}[\/latex]<\/p>\r\n

 <\/p>\r\n

Probabilities can be expressed as decimals, fractions, or percentages.<\/p>\r\n

Notation:<\/strong> The probability of an event is notated as [latex]P(\\text{event})[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n

Adding and subtracting fractions with common denominators: [latex]\\dfrac{a}{c}\\pm \\dfrac{b}{c}=\\dfrac{a\\pm b}{c}[\/latex]
\r\n
\r\nNote that this relationship is described in both directions in the two equations below. That is, it is also true that [latex]\\dfrac{a\\pm b}{c}=\\dfrac{a}{c}\\pm \\dfrac{b}{c}[\/latex]. The second equation furthermore includes the fact that [latex]\\dfrac{a}{a}=1[\/latex].<\/section>\r\n
Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.<\/section>\r\n

In the realm of probability, certain and impossible events represent the extremes of what can occur. An event that cannot happen is deemed impossible, hence it has a probability of [latex]0[\/latex], while an event that is sure to happen is certain, with a probability of [latex]1[\/latex]. All other events fall somewhere between these two extremes, with their probability values reflecting how likely they are to occur.<\/p>\r\n

\r\n
\r\n

certain and impossible events<\/h3>\r\n
    \r\n\t
  • An impossible event<\/strong> has a probability of [latex]0[\/latex].<\/li>\r\n\t
  • A certain event<\/strong> has a probability of [latex]1[\/latex].<\/li>\r\n\t
  • The probability of any event must be [latex]0\\le P(E)\\le 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n
    [ohm2_question hide_question_numbers=1]2808[\/ohm2_question]<\/section>\r\n

    Complement of an Event<\/h2>\r\n

    The concept of the complement of an event in probability is crucial as it helps to understand the likelihood of an event not occurring. By defining the complement, denoted by [latex]\\bar{E}[\/latex] , we can easily calculate its probability by subtracting the probability of the event [latex]E[\/latex] from one. This relationship is fundamental in probability theory, as it connects the occurrence and non-occurrence of an event, completing the picture of all possible outcomes.<\/p>\r\n

    \r\n
    \r\n

    complementary events<\/h3>\r\n

    The complement<\/strong> of an event is the event \u201c[latex]E[\/latex] doesn\u2019t happen\u201d<\/p>\r\n

     <\/p>\r\n
    \r\n

      \r\n\t
    • The notation [latex]\\bar{E}[\/latex] is used for the complement of event [latex]E[\/latex].<\/li>\r\n\t
    • We can compute the probability of the complement using [latex]P\\left({\\bar{E}}\\right)=1-P(E)[\/latex]<\/li>\r\n\t
    • Notice also that [latex]P(E)=1-P\\left({\\bar{E}}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n
      [ohm2_question hide_question_numbers=1]2812[\/ohm2_question]<\/section>","rendered":"

      Basic Probability<\/h2>\n

      The probability of an event is determined by dividing the count of favorable outcomes by the count of all possible outcomes, assuming each outcome has an equal chance of occurring.<\/p>\n

      \n
      \n

      basic probability<\/h3>\n

      Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[\/latex] using this formula:<\/p>\n

       <\/p>\n

      [latex]P(E)=\\frac{\\text{Number of outcomes corresponding to the event E}}{\\text{Total number of equally-likely outcomes}}[\/latex]<\/p>\n

       <\/p>\n

      Probabilities can be expressed as decimals, fractions, or percentages.<\/p>\n

      Notation:<\/strong> The probability of an event is notated as [latex]P(\\text{event})[\/latex]<\/p>\n<\/div>\n<\/section>\n

      Adding and subtracting fractions with common denominators: [latex]\\dfrac{a}{c}\\pm \\dfrac{b}{c}=\\dfrac{a\\pm b}{c}[\/latex]<\/p>\n

      Note that this relationship is described in both directions in the two equations below. That is, it is also true that [latex]\\dfrac{a\\pm b}{c}=\\dfrac{a}{c}\\pm \\dfrac{b}{c}[\/latex]. The second equation furthermore includes the fact that [latex]\\dfrac{a}{a}=1[\/latex].<\/section>\n

      Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.<\/section>\n

      In the realm of probability, certain and impossible events represent the extremes of what can occur. An event that cannot happen is deemed impossible, hence it has a probability of [latex]0[\/latex], while an event that is sure to happen is certain, with a probability of [latex]1[\/latex]. All other events fall somewhere between these two extremes, with their probability values reflecting how likely they are to occur.<\/p>\n

      \n
      \n

      certain and impossible events<\/h3>\n
        \n
      • An impossible event<\/strong> has a probability of [latex]0[\/latex].<\/li>\n
      • A certain event<\/strong> has a probability of [latex]1[\/latex].<\/li>\n
      • The probability of any event must be [latex]0\\le P(E)\\le 1[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n