Algebra 1 Summer Institute 2014

The Fair/Unfair Polarization

Summary
In this activity, participants will determine mathematical and experimental probabilities and compare them to determine if a game is fair or unfair. / Goals
  • Explore finite, equally likely probability models
  • Determine mathematical probabilities and the probability table
/ Participant Handouts
  1. The ESP Polarization

Materials
Paper
Dice / Technology
LCD Projector
Facilitator Laptop / Source
NCTM / Estimated Time
90 minutes

Mathematics Standards

Common Core State Standards for Mathematics
MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models
3.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
3.6: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
3.7: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
  1. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected a random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
  2. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper clip will land open end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies.
3.8: Find the probabilities of compound events using organized lists, tables, trees, and simulation.
  1. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
  2. Represent sample spaces for the compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g.,”rolling double sixes”), identify the outcomes in the sample space which compose the event.
  3. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical experiments
1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
1.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says that a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Standards for Mathematical Practice
  1. Make sense of problems and persevere in solving them
  2. Reason abstractly and quantitatively
  3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use tools appropriately

Instructional Plan

The next activity is a game of chance for two players using two dice of different colors (one red, one blue).

  1. First, what are the mathematical probabilities for a fair die?(Slide 2)

Face / Frequency / Probability
1 / 1 / 1/6
2 / 1 / 1/6
3 / 1 / 1/6
4 / 1 / 1/6
5 / 1 / 1/6
6 / 1 / 1/6
  1. Without them knowing, give them a loaded or unfair die and ask them to roll it 30 times and count the number of times each face shows up.

After they figure out that the die is not fair, give them a fair one so they can repeat the experiment. Pool the results from all participants. How close are the experimental probabilities to the mathematical model?

Suppose you roll a die three times, and the die comes up with a 5 all three times. What is the probability that the fourth roll will be a 5?

  1. Back to the game with two dice. Each of the two players rolls a die, and the winner is determine by the sum of the faces:(Slide 3)
  • Player A wins when the sum is 2, 3, 4, 10, 11, or 12
  • Player B wins when the sum is 5, 6, 7, 8, or 9.

Use your own colored dice to collect data as we play the game.

If this game is played many times, which player do you think will win more often, and why?

For now, let their instincts guide their answer. Later on we'll analyze this problem more thoroughly.

Many people select Player A, since there are more outcomes that will cause this player to win. But in order to be sure, we need to determine the mathematical probability for each player winning. One way to arrive at these mathematical probabilities is to describe all possible outcomes when you toss a pair of dice and compute the sum of their faces.

  1. To analyze this problem effectively, we need a clear enumeration of all possible outcomes. Let's examine one scheme that is based on a familiar idea: an addition table.

Start with a two-dimensional table:

Red Die
+ / 1 / 2 / 3 / 4 / 5 / 6
Blue
Die / 1
2
3
4
5
6

Ask participants to complete the table (be aware of the difference between such outcomes as 2 + 4 and 4 + 2 since they come from different dice. How many entries does the table have?

  1. For how many of the 36 outcomes will Player A win? For how many of the 36 outcomes will Player B win? Who is more likely to win this game?
  1. Change the rules of the game in some way that makes it equally likely for Player A or Player B to win.

One potential change is to change the sums that each players wins with. Here's one possible solution:

• / Player A wins when the sum is 2, 3, 4, 7, 10, 11, or 12.
• / Player B wins when the sum is 5, 6, 8, or 9.

It may seem surprising that this is a fair game, but with this change each player will win one-half (18/36) of the time.

  1. Another way to solve this problem is to look at a probability table for the sum of the two dice. This representation can be quite useful, since it gives us a complete description of the probabilities for the different values of the sum of two dice, independent of the rules of the game.

Using the table of the sums, ask participants to complete the probability table:

Sum / Frequency / Probability
2 / 1 / 1/36
3 / 2 / 2/36
4 / 3 / 3/36
5 / 4 / 4/36
6 / 5 / 5/36
7 / 6 / 6/36
8 / 5 / 5/36
9 / 4 / 4/36
10 / 3 / 3/36
11 / 2 / 2/36
12 / 1 / 1/36
  1. Use the probability table you completed to determine the probability that Player A will win the game. Recall that Player A wins if the sum is 2, 3, 4, 10, 11, or 12.

This can be found by adding the probabilities of the sums that Player A will win with:

1/36 + 2/36 + 3/36 + 3/36 + 2/36 + 1/36 = 12/36

Player A wins with probability 12/36, or one-third of the time.

  1. If we know the probability that Player A wins, how could you use it to determine the probability that Player B wins without adding the remaining values in the table?(Slide 5)

Since one of the two players has to win, the sum of both probabilities -- that of Player A winning and that of Player B winning -- is 1. So a faster way to find the probability that Player B wins is to subtract the probability that Player A wins (12/36) from 1:

1 - (12/36) = (36/36) - (12/36) = 24/36

Player B wins with probability 24/36, or two-thirds of the time.

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