Lesson 8: How Likely Is It?

Objective:

Students will understand the basic rules of probability. They will learn that previous outcomes do not give information about future outcomes if the events are independent.

Materials:

Video: “Heads” from the movie Rosencrantz and Guildenstern are Dead found at:
https://www.youtube.com/watch?v=NbInZ5oJ0bc

https://www.youtube.com/embed/NbInZ5oJ0bc
Projector for RStudio functions

Vocabulary:

probability simulation model sample proportion chance independence

Essential Concepts:

Probability is an area about which we humans have poor intuition. Probability measures a long-run proportion: 50% chance means the event happens 50% of the time if you repeated it forever. When we don't repeat forever, we see variability.

Lesson:

Ask students to consider possible synonyms for the word chance. If someone says, “That just happened by chance,” what does that mean? Synonyms: possibility, prospect, expectation, unintentional, unplanned. The actual definition of chance is “a possibility of something happening.”
Then, ask them which game – chess or the board game, “Sorry” – is more based on chance. Why?
Note: Any game can be chosen. "Sorry" is more based on chance because many outcomes are determined by dice rolls. In chess there are certain strategies and movements that can be planned, so it is more a game of skill. For "Sorry" the players roll a die (number cube), so the numbers they roll have an impact on how well they do in the game.
Next ask students if they can think of situations where chance is the only force at play. Possible responses: card games, slot machines, the lottery, coin flipping, and rock-paper-scissors.
Play the “Heads” video from the movie Rosencrantz and Guildenstern are Dead found at: https://www.youtube.com/watch?v=NbInZ5oJ0bc.
In their IDS Journals, ask students to write down their initial reactions to the clip by responding to the following questions:
1. Is it possible to get 78 heads in a row when tossing a coin? Yes, it is possible to get 78 heads in a row since one coin toss does not determine the next coin toss.
2. Do you think it is likely to get 78 heads in a row? No, although it is possible to get 78 heads in a row.
3. How many times should we get heads when tossing a coin? 1 out of 2 times, or 50% of the time.
4. On average, how many times out of the 78 tosses should the characters have gotten heads? Roughly 39 times.
Ask students to discuss their findings with their team members and come to an agreement on their responses. Afterwards, conduct a Whip Around and ask each team to share its findings. Are there any differences between the teams? Any similarities?
As teams share their responses, students should add to or revise their individual findings in their IDS Journals.
Explain to students that, from the concept of chance, we can start learning about probability. Chance is simply the possibility that something will happen, and probability is a measurement for how often something happens in the “long run.” Students may have ideas about how to calculate probabilities based on prior classes or knowledge, but inform them that IDS will be taking a different approach by using simulations (see next step).
Since we don’t want to actually flip a coin 78 times like the actors did in the video, we can have RStudio simulate them for us. A simulation is a way of creating random events that are close to real-life situations without actually doing them. It is a kind of model, which is a way of representing real-world situations so that predictions can be made.
Explain to students that R has a function that does coin flipping for us, and that it assumes an equal probability of heads and tails. Using a projector to display your computer screen to the whole class, demonstrate how to do one simulation of a coin flip in RStudio. Use the following function:

> rflip(1)
Explain that the value of 1 in the argument part of the function tells R to flip the coin 1 time. If we want to flip the coin 10 times, we could simply change the function to rflip(10).
Run the function again using 10 as the number of times to flip the coin. Ask students:
1. How many heads (“H”s) were there? Answers will vary for each sample.
2. How many Tails (“T”s) were there? Answers will vary for each sample.
3. In the output, what does Flipping 10 coins [Prob(Heads) = 0.5] mean? This is RStudio telling us that we are tossing the coin 10 times and that the probability of getting heads is 0.5 because it is a fair coin.
4. In the output, what does Number of Heads: 3 [Proportion Heads: 0.3] mean?
  
  Note: This is an example of an output. Your sample may have a different value for the number of heads that appeared, and therefore a different value for the proportion of heads. This is RStudio telling us that in our sample, we got heads 3 out of the 10 times we flipped the coin. The sample proportion is automatically calculated for us by dividing the number of heads by the total number of tosses (in this case, 3/10 = 0.3).
To relate back to the video at the beginning of class, repeat the simulation once more, but use 78 as the number of coin flips rflip(78). Ask students:
1. How many heads (“H”s) were there? Since we know to expect about 39 heads if the coin is fair, does the value seem reasonable? Answers will vary for each sample. Most likely, you will see values near 39.
2. How many Tails (“T”s) were there? Answers will vary for each sample.
3. What proportion of the coin flips were heads? Answers will vary for each sample.
Using the rflip(78) command, run the simulation 3-5 more times and have students record the values for the number of heads and the proportion of heads.

As an example, we ran the function 3 times and saw the following values:

Sample 1 – amount of heads: 45

        proportion of heads: 0.577

Sample 2 – amount of heads: 33

        proportion of heads: 0.423

Sample 3 – amount of heads: 42

        proportion of heads: 0.538
Have students answer the questions below. The important thing to note is that the values can and (almost always) WILL change each time you run the simulation to create a new sample.
1. How do the proportions of heads in the samples compare to each other? Answers will vary.
2. How do the proportions of heads compare to the true probability of heads (1/2 or 50%)? Answers will vary, but students should notice that most of the probabilities are close to 50%.
3. Why is there a 50% chance of getting heads during each coin flip? Since there are two sides to a coin, both should be equally likely to come up. So there is a 1 out of 2 chance of getting heads and 1 out of 2 chance of getting tails.
Ask students to engage in a discussion with their group about the statement below, then have a few group reporters share out.
1. If a coin was flipped 78 times, I would claim that the coin is unfair if I got less than # heads or more than # heads.
Inform students that you are going to perform 500 simulations. Each simulation represents a coin being flipped 78 times. For each simulation, the computer will record the number of heads in the 78 flips. A histogram will be created that represents the number of heads in each of the simulations. The histogram is a model that will display what typically happens when a fair coin is flipped 78 times.

Copy and paste the code below in an RScript and run each line of code, one at a time, for the students:

set.seed(11) #reproducibility 
flips <- do(500)*rflip(78)  
View(flips) # 4 variables
histogram(~heads, data = flips)  
favstats(~heads, data = flips)

Engage the students in a discussion about the histogram:
1. What is this distribution telling us? When flipping a fair coin 78 times, what typically happened was that it landed on heads between 36 and 40 times (36/78 = 0.46 to 40/78 = 0.51). It was not uncommon for the coin to land on heads 31-35 (0.40-0.45) times or 41-45 (0.52-0.58) times. Even landing on heads between 46-50 (0.59-0.64) times was not too uncommon. What was very uncommon, however, was landing on heads less than 30 times (less than 38%) or more than 51 times (more than 65%).
2. Were your group's cut-offs (item #16) similar to what the chance model displayed? Answers will vary. Some groups' intervals might be very wide and others very narrow.
3. Using the chance model (histogram) which displays what typically happens when a fair coin is flipped 78 times, make a call for the scenarios below - fair or unfair?
  1. You flip a coin 78 times and get 37 heads. Fair. 37 was very common based on the histogram.
  2. You flip a coin 78 times and get 46 heads. Fair. 46 was less common but not too uncommon.
  3. You flip a coin 78 times and get 20 heads. Unfair. In the 500 simulations not once did we see a FAIR coin land on heads 20 times.
Next pose the following question:
1. If you get a heads on the first toss of a coin, will you definitely get a heads on the next toss? Will you definitely get a tails on the next toss? No. One coin toss should not affect another coin toss. Each time you flip the coin, the chances of getting heads versus tails remains the same.
Introduce the concept of independence. Explain that, when tossing a fair coin, there is no relationship between each toss. The second toss does NOT depend on the first toss; therefore, the coin tosses are independent of each other.

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

Homework

Students will create a Tweet (they do not have to post it online). Using 280 characters or fewer, write a Tweet about the meaning of probability.