Lesson 4

The purpose of this warm-up is for students to practice placing numbers represented with decimals on a number line and thinking about probabilities of events that involve the values of the numbers. In the following activity, students are asked to graph points involving probabilities that are represented by numbers similar to the ones in this activity.

Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by time to share their responses with a partner. Follow with a whole-class discussion. 

Select some partners to share their responses and methods for positioning the points on the number line. If time allows, select students to share a chance event for each of the values listed.

This activity begins to answer the question brought up in the previous lesson about finding the probability when the sample space is not available. Students have the opportunity to use this experiment for which the sample space is available to check its agreement and estimate based on repeating the experiment many times (MP8).

Students make the connection between probability and the fraction of outcomes for which the event occurs in the long-run. This activity highlights that a probability describes what happens in the the long run and that it does not guarantee that the event will occur a specific number of times after any specific number of trials. For example, an event that has probability 0.6 means that the event will occur about 60% of the time in the long run, but it does not mean that it will occur exactly 60 times when the experiment is performed 100 times.

Students may not notice a pattern in the graph. Ask if they can see a pattern with the decimal values for the fraction of wins in their table. If their data does not fit the expected pattern, tell them that this is not typical and ask them to look at another group's results.

The purpose of this discussion is for students to understand that computing the fraction of the time an event occurs can be used to estimate the probability of the event and that more repetitions should make the estimation more accurate. 

Select some students to share their answer and reasoning for the second question. If it is not mentioned by students, tell them that when there is more than one outcome that is in the desired event, then the probability of that event is the number of outcomes in the desired event divided by the number of outcomes in the sample space. In this example, there are 2 outcomes that win (a roll of 1 or 2) and 6 outcomes in the sample space, so the probability of winning is \(\frac{2}{6}\) which is equivalent to \(\frac{1}{3}\).

Collect the number of 1s and 2s for each group and compute the fraction for the whole class with all the data. The value should be very close to \(\frac{1}{3}\).

Select students to share their thoughts on what appears to be happening with the points on their graph. (They are leveling out at 0.33.) If students struggle with noticing that the points are leveling out at a \(y\) value of 0.33, ask them to draw a horizontal line on their graphs at their answer for the probability they got in the second question. 

Ask the class how many times the entire class rolled number cubes. Then ask, "Based on the probability predicted in the second question, how many times do we expect the class to have simulated a win for Mai? How does this compare to the actual number of wins the class rolled."

A probability tells you how likely an event is to occur. While it is not guaranteed to be an exact match, if the chance experiment is repeated many times, we expect the fraction of times that an event occurs to be fairly close to the calculated probability.

This activity gives students the opportunity to see that an estimate of the probability for an event should be close to what is expected from the exact probability in the long-run; however, the outcome for a chance event is not guaranteed and estimates of the probability for an event using short-term results will not usually match the actual probability exactly (MP6).

Tell students that the probability of a coin landing heads up after a flip is \(\frac{1}{2}\). 

Give students 5 minutes of quiet work time followed by a whole-class discussion.

The purpose of the discussion is for students to recognize that the actual results from repeating an experiment should be close to the expected probability, but may not match exactly.

For the first problem, ask students to indicate whether or not they think each result seems surprising. For the second and third questions, select several students to provide answers and display for all to see, then create a range of values that might not be surprising based on student responses. Ask the class if they agree with this range or to provide a reason the range is too large. It is not important for the class to get exact values, but a general agreement should arise that some range of values makes sense so that there does not need to be exactly 50 heads from the 100 flips.

An interesting problem in statistics is trying to define when things get "surprising." Flipping a fair coin 100 times and getting 55 heads should not be surprising, but getting either 5 or 95 heads probably is. (Although there is not a definite answer for this, a deeper study of statistics using additional concepts in high school or college can provide more information to help choose a good range of values.)

Explain that a probability represents the expected likelihood of an event occurring for a single trial of an experiment. Regardless of what has come before, each coin flip should still be equally likely to lands heads up as tails up.

As another example: A basketball player who tends to make 75% of his free throw shots will probably make about \(\frac{3}{4}\) of the free throws he attempts, but there is no guarantee he will make any individual shot even if he has missed a few in a row.