Lesson 5

The purpose of this warm-up is for students to think more deeply about probabilities and what the values actually represent. In this activity, students are asked to compare the likelihood of three events with probabilities given in different formats. In the discussion, students are also asked to think about the context of the situations to see that probabilities are not the only consideration when planning a response.

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Ask students, "Which situation would you worry about the most? Is that the same situation that is the most likely?"

Note that our interpretation of the scenario influences how we feel about how likely an event is to happen. Although the likelihood of rain is higher, the implications of a tornado are much greater, so you may be more likely to worry about the tornado than the rain.

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.

Students may think they need to have their probability estimates match the computed probability. Remind them of the activity "Due For a Win" to see why we might expect the estimated probability and the calculated probability to be a little different. 

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 fraction of 3 spins compare to the probability you expect from just looking at the spinner?"

Explain that although the spinners provided were designed to have equally sized sections (except for spinner D which has the angles \(180^\circ, 45^\circ,\) and \(90^\circ\)), sometimes it may be difficult to determine when the sections are exactly the same size. For some 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.
  • Maybe the spinner sections were not equal sized when they should be or maybe the spinner was tilted.
  • Maybe the items being chosen from the bag were different sizes, so you were more likely to grab one than another.
  • Maybe the coin or number cube are not evenly weighted and are more likely to land with one side up than others.
  • Maybe the computer that was programmed to return a random number has a problem with the code that returns some numbers more often than others.
• Not enough trials were run.
  • As in the previous lesson, if you flip a coin once and it comes up heads, that doesn’t mean that it always will. Even if you flip it 100 times, it’s not guaranteed to land heads up exactly 50 times, so some slight deviation is to be expected.

In the previous activity, students could see the entire spinner and compute the probability to compare with their experimental results. In many contexts, it is not possible to know the entire sample space or compute an exact probability from the situation itself. For example, the probability of rain tomorrow usually cannot be exactly estimated from available information. In this activity, students see how to approach such problems 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 we have available. Students gain exposure to the process of drawing blocks from a bag, which will be useful in designing simulations in future lessons. 

Arrange students in groups of 3–4. Distribute 1 paper bag containing 5 snap cubes (3 green cubes and 2 cubes of some other color that match each other) to each group. 5 minutes of group work followed by a whole-class discussion.

Some students may estimate a probability that is different from the fraction of times they draw a green block. Ask these students for a 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, since not everyone picked out the same thing each time.)
• "Some of you may have felt that there are 5 blocks in the bag. If we use that information, does that change our estimate of the probability?"  (If there are only 5 blocks, it only makes sense for the probability to be \(0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5},\) or \(\frac{5}{5}\).)
• Allow students to open the bags and see what blocks are in there. "What is the probability based on this observation?" (\(\frac{3}{5}\))
• "Does it match your group estimates? Does it match the class estimates?"
• "Was the estimate from the class data more accurate than the estimates from the groups?" (It should be since more trials are included.)