## The Law of Averages

Instructions:
• Use the radio buttons to choose to toss a coin or roll a die.
• An experiment consists of 10000 trials (tosses or rolls).
• Press the 'Get data' button to conduct an experiment.
• By default, the applet plots the percentage of successes (a success is a 'head' for the coin or a 'three' for the die) by the number of trials.
• Up to 25 experiments can be overlayed on the same plot.
• The current or newest experiment is plotted in orange; previous experiments are shown in a default plot color.
• The radio buttons at the bottom of the applet allow you to choose to plot the percentage of successes minus the expected percentage of successes or the absolute number of successes minus the expected number of successes.
• Use the 'reset' button to clear the plot.

In illustration of the law of averages, you will observe that as the number of trials increases, the percentage of successes approaches the expected percentage of successes while the difference between the observed and expected number of successes increases.

Scale - Note that the scale on the horizontal axis is not constant. The plots may appear to show sudden jumps or dramatic differences at 100 and 1000 tosses, this is a result of the change of scale at those positions.

Display – The 'Display' menu in the upper left hand corner of the applet allows the user to adjust the speed of the experiments and change the appearance of the plot.

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Coin Activity

Select the 'coin' radio button on the applet. Press the 'Get data' button to toss the coin 10000 times, do this several times so the results of a number of experiments are overlain in the plot. Use the plots to respond to the prompts below.
1. Move between the plot display radio buttons ('show %heads' et al.) as needed to answer the following questions.

• Are you more likely to get more than 60% heads in 10 tosses or 1000 tosses of a coin? Explain.
• Are you more likely to get more than 40% heads in 10 tosses or 1000 tosses of a coin? Explain.
• Are you more likely to get between 60% and 40% heads in 10 tosses or 1000 tosses of a coin? Explain.

2. Are you more likely to get exactly 50% heads in 10 tosses or 1000 tosses of a coin? Explain.

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Select the 'die' radio button on the applet. Press the 'Get data' button to roll the die 10000 times, do this several times so the results of a number of experiments are overlain in the plot. Use the plots to respond to the prompts below.
1. Move between the plot display radio buttons ('show %threes' et al.) as needed to answer the following questions.

• Are you more likely to have at least 10 more than the expected number of threes for 100 rolls or 1000 rolls?
• Are you more likely to be within 5% of the expected percentage of threes for 100 rolls or 1000 rolls?

2. Are you more likely to get exactly the expected percentage of threes in 10 rolls or 1000 rolls of a die? Explain.
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