In this activity, students return to calculating probabilities using the sample space, and they compare the calculated probabilities to the outcomes of their actual trials. Students have a chance to construct arguments (MP3) about why probability estimates based on carrying out the experiment many times might differ from the expected probability. Students use a spinner in this activity, which will be helpful when designing simulations in upcoming lessons.

In the digital version of the activity, students use an applet to spin the spinners virtually. The applet allows students to focus on the mathematical goals rather than creating and using the spinners themselves. The digital version may be preferable if students do not need or want tactile interaction with physical objects.

The purpose of this discussion is to think about reasons why the estimate of a probability may be different from the actual probability.

Select some students to share their responses to the last 5 questions. 

Ask, “How does your estimate for the probability of spinning a 3 compare to the probability you expect from just looking at the spinner?”

Explain that although the spinners provided are designed to have equally sized sections (except for Spinner D which has the angles 180 degrees, 45 degrees, and 90 degrees), sometimes it may be difficult to determine when the sections are exactly the same size. For situations where things are not so evenly divided, some experimenting may be needed to determine that the outcomes actually follow the probability we might expect.

There are two main reasons why the fraction of the time an event occurs may differ from the actual probability:

• The simulation was designed or run poorly. For example:
  • Maybe the spinner sections were not of equal size even though they should have been, or maybe the spinner was tilted.
  • Maybe the items being chosen from the bag were different sizes, so it was more likely to grab one item than another.
  • Maybe the coin or number cube were not evenly weighted and were more likely to land with one side up than another.
  • Maybe the computer that was programmed to return a random number had a problem with the code, making it return some numbers more often than others.
• Not enough trials were run. For example:
  • As in the previous lesson, if a coin were flipped once and it came up heads, that doesn’t mean that it always will. Even if it were flipped 100 times, it’s not guaranteed to land heads up exactly 50 times. Some slight deviation is to be expected.

In this activity, students see how to approach understanding probability when the entire sample space cannot be known or if the situation is more complex by estimating the probability of an event using the results from repeating trials (MP8). In this particular example, the exact probability can be computed when the information is revealed, so students can compare their results to this value. In situations like predicting the weather, estimates may be the best thing available. Students gain exposure to the process of drawing blocks from a bag, which will be useful in designing simulations in future lessons. 

Some students may estimate a probability that is different from the fraction of times they draw a green block. Ask these students for the reason they chose a different value for their estimate. 

The purpose of the discussion is to show that estimating the probability of an event can be done using repeated trials and is usually improved by including more trials.

Ask each group how many green blocks they got in their trials and display the class results for all to see.

Consider asking these discussion questions:

• “How can we use the values from the class to estimate the probability of drawing out a green block?” (By using the data we have, we can estimate the fraction of blocks that are green.)
• “Based on the class data, what is the estimated probability of choosing a green block from the bag?”
• “Was the probability estimated from the class data different from the probability estimate based on the data just from your group? Why?” (Yes, it was different. Not everyone picked out the same thing each time, and in the class data there were more trials.)
• “Some of you may have guessed that there are 5 blocks in the bag. If we use that information, does it change our estimate of the probability?” (If there are only 5 blocks, it only makes sense for the probability to be or .)
• Allow students to open the bags and see what blocks are in there. “What is the probability based on this observation?” ()
• “Does it match your group’s estimate? Does it match the class estimate?”
• “Was the estimate from the class data more accurate than the estimates from the groups?” (It should be more accurate since more trials are included.)