{"id":340,"date":"2020-07-08T14:15:59","date_gmt":"2020-07-08T18:15:59","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=340"},"modified":"2025-04-25T18:27:25","modified_gmt":"2025-04-25T22:27:25","slug":"3-2-probability-of-an-event","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/3-2-probability-of-an-event\/","title":{"raw":"3.2 Probability of An Event","rendered":"3.2 Probability of An Event"},"content":{"raw":"Given an event [latex]E[\/latex], the chance or the probability that the event [latex] E[\/latex] happens is denoted as [latex] P(E)[\/latex]. A probability near 0 indicates that the event is very unlikely to occur when the chance experiment is conducted, whereas a probability near 1 suggests that the event is very likely to occur.\r\n

3.2.1 Frequentist Interpretation of Probability<\/strong><\/h2>\r\nFrom the frequentist\u2019s point of view, the probability of an event can be interpreted as the proportion of times the event occurs in a large number of repetitions of the chance experiment. For instance, if we flip a balanced coin 100 times and observe 55 heads, then the probability of observing a head is\r\n

[latex] P(H) \\approx \\frac{\\text{\\# of times we observe a head}}{n} = \\frac{55}{100} = 0.55.[\/latex]<\/p>\r\nThe figure below shows that the proportion approaches to 0.5 as the number of experiments repetitions [latex] n[\/latex] increases. It is expected to observe heads half of the time, because the coin is balanced and there is a 50\u201350 chance heads are observed. Therefore, the proportion of observed heads will approach a constant when the coin is flipped infinite times, this constant is [latex]P(H).[\/latex]<\/a>\r\n\r\n[caption id=\"attachment_2686\" align=\"aligncenter\" width=\"480\"]\"A Figure 3.1<\/strong>: Proportion of Heads Converges to 0.5. [Image Description <\/a>(See Appendix D Figure 3.1)<\/a>][\/caption]Theoretically speaking, it is impossible to observe the probability of an event [latex] P(E)[\/latex], because we won\u2019t repeat the chance experiment infinite times. However, we can sometimes calculate the probability based on a model or some probability rules.\r\n

3.2.2 Equally Likely Outcome Model, the f\/N Rule<\/h2>\r\nThe simplest model, the equally likely outcome model, assumes that all possible outcomes have equal chance to be observed, such as flipping a balanced<\/strong> coin and rolling a fair <\/strong>die. For the equally likely outcome model, the probability that an event E happens is given by\r\n\r\n[latex]\\begin{align*}\r\nP(E) &= \\frac{\\text{\\# of sample points in event }E}{\\text{\\# of sample points in sample space }S} \\\\\r\n&= \\frac{\\text{\\# of ways event E can occur}}{\\text{\\# of possible outcomes}} \\\\\r\n&= \\frac{f}{N}.\r\n\\end{align*}[\/latex]\r\n
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Example: Equally Likely Outcomes<\/p>\r\n\r\n<\/header>\r\n

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    \r\n \t
  1. Recall that a standard die has sample space\u00a0[latex] S[\/latex]= {1, 2, 3, 4, 5, 6}. Use the equally likely outcomes model to find the probability of the following events:<\/li>\r\n<\/ol>\r\n
      \r\n \t
    1. Observing a six: E = {6}\r\n

      [latex]P(E) = \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample in sample space S}} = \\frac{f}{N} = \\frac{1}{6}.[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n

        \r\n \t
      1. Observing an even number: A = {2, 4, 6}<\/li>\r\n<\/ol>\r\n

        [latex] P(A) = \\frac{\\text{\\# of sample points in event A}}{\\text{\\# of sample points in space S}} = \\frac{f}{N} = \\frac{3}{6}=\\frac{1}{2}.[\/latex]<\/p>\r\n\r\n

          \r\n \t
        1. Observing an outcome that is less than 3: B = {1, 2}<\/li>\r\n<\/ol>\r\n

          [latex] P(B) = \\frac{\\text{\\# of sample points in event B}}{\\text{\\# of sample points in space S}} = \\frac{f}{N} = \\frac{2}{6} = \\frac{2}{3}.[\/latex]<\/p>\r\n\r\n

            \r\n \t
          1. Suppose there are 100 students in a class, the following table summarizes the frequencies of number of siblings the students have:<\/li>\r\n<\/ol>\r\n

            Table 3.2<\/strong>: Frequency and Relative Frequency of # of Siblings<\/p>\r\n\r\n

            \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            Total<\/td>\r\n100<\/td>\r\n1.00<\/td>\r\n<\/tr>\r\n<\/tfoot>\r\n
            # of<\/strong> Siblings<\/strong><\/td>\r\nFrequency<\/strong><\/td>\r\nRelative Frequency<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
            0<\/td>\r\n10<\/td>\r\n0.10<\/td>\r\n<\/tr>\r\n
            1<\/td>\r\n30<\/td>\r\n0.30<\/td>\r\n<\/tr>\r\n
            2<\/td>\r\n35<\/td>\r\n0.35<\/td>\r\n<\/tr>\r\n
            3<\/td>\r\n15<\/td>\r\n0.15<\/td>\r\n<\/tr>\r\n
            >3<\/td>\r\n10<\/td>\r\n0.10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

            Find the probability that a randomly selected student has:<\/p>\r\n\r\n

              \r\n \t
            1. exactly one sibling<\/li>\r\n<\/ol>\r\n

              If we randomly pick one student, each student has the same chance to be chosen; therefore, we can use the equally likely outcome model. There are [latex]\u00a0f=30[\/latex] students with exactly one sibling. Hence<\/p>\r\n

              [latex] P(E) = \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample points in sample space S}} = \\frac{f}{N} = \\frac{30}{100} = 0.3.[\/latex]<\/p>\r\n\r\n

                \r\n \t
              1. at least one sibling, which means one, two, three, or more than three siblings.<\/li>\r\n<\/ol>\r\n

                [latex]P(E) = \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample points in sample space S}} = \\frac{f}{N} = \\frac{30+35+15+10}{100} = 0.9.[\/latex]<\/p>\r\n\r\n

                  \r\n \t
                1. two to three siblings (inclusive), which means either two or three siblings.<\/li>\r\n<\/ol>\r\n

                  [latex] P(E) = \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample points in sample space S}} = \\frac{f}{N} = \\frac{35+15}{100} = 0.5.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                  \r\n

                  Key Facts: Basic Properties of the Probability of an Event<\/p>\r\n\r\n<\/header>\r\n

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