{"id":30,"date":"2026-01-25T00:27:33","date_gmt":"2026-01-25T00:27:33","guid":{"rendered":"https:\/\/learnx.ca\/um\/?page_id=30"},"modified":"2026-02-16T21:26:08","modified_gmt":"2026-02-16T21:26:08","slug":"probability","status":"publish","type":"page","link":"https:\/\/imaginethis.ca\/u\/probability\/","title":{"rendered":"Probability"},"content":{"rendered":"\n

\u201cI believe that we do not know anything for certain, but everything probably.\u201d
\u2015Christiaan Huygens<\/p>\n\n\n\n

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Binary choice<\/h1>\n\n\n\n

What do you see in the image below? What conceptual connections can you make?<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
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TOSSING 1 COIN<\/h2>\n\n\n\n

Imagine doing this:<\/p>\n\n\n\n

    \n
  • Flip a coin.<\/li>\n\n\n\n
  • If it’s Heads<\/strong>, take one step along the red path<\/strong>.<\/li>\n\n\n\n
  • If it’s Tails<\/strong>, take one step along the yellow path<\/strong>.<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    Which result will more likely? A or B?<\/p>\n\n\n\n

    If you did this experiment several times and kept a tally, it might look like this.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    The 2 outcomes are equally likely.<\/p>\n\n\n\n

    \n\n\n\n

    TOSSING 2 COINS<\/h3>\n\n\n\n

    Imagine doing this:<\/p>\n\n\n\n

      \n
    • Flip a coin.\n
        \n
      • If it’s Heads<\/strong>, take one step along the red path<\/strong>.<\/li>\n\n\n\n
      • If it’s Tails<\/strong>, take one step along the yellow path<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Flip the coin a second time.\n
          \n
        • If it’s Heads<\/strong>, take one step along the red path<\/strong>.<\/li>\n\n\n\n
        • If it’s Tails<\/strong>, take one step along the yellow path<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
          \"\"<\/figure>\n\n\n\n

          Which result is more likely\u2014A, B, or C?<\/p>\n\n\n\n

          Most young children (Grades 1\u20132) initially believe all three outcomes are equally likely.
          When they carry out the experiment, they are surprised to discover that B occurs twice as often<\/strong>.<\/p>\n\n\n\n

          Interestingly, some children can already explain why: they notice that there are two paths leading to B<\/strong>, but only one path each<\/strong> leading to A and to C.<\/p>\n\n\n\n

          This simple paths diagram<\/em> provides children (and adults) with a powerful cognitive scaffold\u2014offering clarity, reducing complexity, and supporting the development of deeper conceptual understanding.<\/p>\n\n\n\n

          \n\n\n\n

          TOSSING A COIN 5 TIMES<\/h3>\n\n\n\n

          Now imagine flipping a coin 5 times and travelling left of right based on Heads or Tails.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Which result will more likely? A, B, C, D, E or F?<\/p>\n\n\n\n

            \n
          • Which letters are more likely to win? <\/li>\n\n\n\n
          • Which letters are more likely to lose? <\/li>\n\n\n\n
          • Why?<\/li>\n<\/ul>\n\n\n\n
            \n\n\n\n

            PASCAL’S TRIANGLE<\/h3>\n\n\n\n

            How do the paths patterns connect to Pascal’s triangle?<\/p>\n\n\n\n

            \"\"<\/figure>\n\n\n\n

            Did you know that this pattern was well known in China, several centuries before Pascal?<\/p>\n\n\n\n

            It was know was Yang Hui’s triangle.<\/strong><\/p>\n\n\n\n

            \"\"
            Image from Wikipedia<\/figcaption><\/figure>\n\n\n\n
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            CONNECTIONS ACROSS REPRESENTATIONS<\/h3>\n\n\n\n

            What connections do you notice?<\/p>\n\n\n\n

            \"\"<\/figure>\n\n\n\n
            \n\n\n\n

            BINARY CHOICE<\/h3>\n\n\n
            \n
            \"\"<\/figure>\n<\/div>\n\n\n

            When you toss a coin, you\u2019re making a binary choice<\/strong>\u2014a choice between just two options: heads or tails<\/strong>.<\/p>\n\n\n\n

            A light switch works the same way. It offers a simple binary state: on or off<\/strong>.<\/p>\n\n\n\n

            The idea of binary choice is remarkably powerful.
            It appears in probability<\/strong> (coin tossing), in algebra<\/strong> (expressions like x + y<\/em> that represent two possibilities), and in the binomial theorem<\/strong>, which beautifully links probability with algebra by showing how multiple binary events combine and expand.<\/p>\n\n\n\n

            Digital devices, like smartphones and computers, rely on countless combinations of these binary states. Their internal circuits operate using 1 (on)<\/strong> and 0 (off)<\/strong>, making the binary number system<\/strong> perfectly suited to how they process information.<\/p>\n\n\n\n

            \"\"
            Binomial Theorem<\/figcaption><\/figure>\n\n\n\n
            \n\n\n\n

            STATISTICIAN INTERVIEW<\/h3>\n\n\n\n

            Interview with statistician Dr. Bethany White (University of Toronto).<\/p>\n\n\n\n