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Evaluating Statements about Probability
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Homework
  • Improving Individual Solutions to the Assessment Task (10 minutes)
      • Return the original assessment task, Are They Correct?, together with a blank copy of the task.
        • Look at your original responses, and think about what you have learned in this lesson.
        • Using what you have learned, try to improve your work.
      • If you have not added questions to individual pieces of work, then write your list of questions on the board.
      • Students should select from this list only the questions they think are appropriate for their own work.
  • Solutions: Assessment Task: Are They Correct?
      1. This statement is incorrect. It highlights the misconception that all events are equally likely. There are many factors (e.g., the season) that will influence the chances of it raining tomorrow.
      2. Assuming that the sex of a baby is a random, independent event equivalent to tossing a coin, the statement is incorrect. It highlights the misconception that later random events can "compensate" for earlier ones. The assumption is important: there are many beliefs and anecdotes about what determines the sex of a baby, but "tossing a coin" turns out to be a reasonably good model¹.
      3. This statement is incorrect. This highlights the misconception that "special" events are less likely than "more representative" events.

      ¹ See, for example, http://www.bbc.co.uk/news/magazine-12140065

  • Solutions: Collaborative Activity: True, False, or Unsure
      • A: If you roll a six-sided number cube, and it lands on a six more than any other number, then the number cube must be biased.

        False. This statement addresses the misconception that probabilities give the proportion of outcomes that will occur. With more information (How many times was the cube rolled? How many more sixes were thrown?), more advanced mathematics could be used to calculate the probability that the dice was biased, but you could never be 100 percent certain.

      • B: When randomly selecting four letters from the alphabet, you are more likely to come up with D, T, M, J than W, X, Y, Z.

        False. This highlights the misconception that "special" events are less likely than "more representative" events. Students often assume that selecting the "unusual" letters W, X, Y and X is less likely.

      • C: If you toss a fair coin five times and get five heads in a row, the next time you toss the coin it is more likely to show a tail than a head.

        False. This highlights the misconception that later random events "compensate" for earlier ones. The statement implies that the coin has some sort of "memory." People often use the phrase "the law of averages" in this way.

      • D: There are three outcomes in a soccer match: win, lose, or draw. The probability of winning is therefore ⅓.

        False. This highlights the misconception that all outcomes are equally likely, without considering that some are much more likely than others. The probabilities are dependent on the rules of the game and which teams are playing.

      • E: When two coins are tossed there are three possible outcomes: two heads, one head, or no heads. The probability of two heads is therefore ⅓.

        False. This highlights the misconception that all outcomes are equally likely, without considering that some are much more likely than others. There are four equally likely outcomes: HH, HT, TH, TT. The probability of two heads is ¼.

      • F: Scoring a total of three with two number cubes is twice as likely as scoring a total of two.

        True. This highlights the misconception that the two outcomes are equally likely. To score three there are two outcomes, 1,2 and 2,1, but to score two there is only one outcome, 1,1.

      • G: In a "true or false?" quiz with ten questions, you are certain to get five correct if you just guess.

        False. This highlights the misconception that probabilities give the exact proportion of outcomes that will occur. If a lot of people took the quiz, you would expect the mean score to be about 5, but the individual scores would vary.

        Probabilities do not say for certain what will happen, they only give an indication of the likelihood of something happening. The only time we can be certain of something is when the probability is 0 or 1.

      • H: The probability of getting exactly two heads in four coin tosses is ½.

        False. This highlights the misconception that the sample size is irrelevant. Students often assume that because the probability of one head in two coin tosses is ½, then the probability of n heads in 2n coin tosses is also ½. In fact the probability of two out of four coin tosses being heads is:

        6
        16

        This can be worked out by writing out all the sixteen possible outcomes:

        HHHH, HHHT, HHTH, HTHH, THHH, TTTT, TTTH, TTHT, THTT, HTTT, HHTT, HTTH, TTHH, THTH, HTHT, THHT.

        This may be calculated from Pascal's triangle:

        Student are not expected to make this connection!

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