Dice Sum Bingo
In this activity, students design bingo cards based on their impressions of which sums will come up first and most often when two dice are rolled repeatedly. This activity is a nice introduction to discrete probability distributions and some of their conceptual pitfalls.
Setup
All of the students should draw a \(3 \times 3\) grid on a sheet of paper. The students can fill out the nine spaces of this grid with nine numbers of their choosing, and can even use numbers more than once, but they might find some numberings more helpful to win the game they're about to play! The way the game works is:
The facilitator will roll two standard \(6\)-sided dice and add the two results, calling out this sum.
If a sum is called and it is on a student's card, they may cross it off. They may only cross off one copy of it. (If a player's card has three instances of \(5\) and a \(5\) is called, they may cross off one of their \(5\)'s and then must wait for \(5\) to be called again to cross off another.)
The first player to cross off all nine numbers on their card wins.
Initial Questions
What numbers does it make sense to put on the card? Are there any numbers that you shouldn't put on your card? (For example, does \(1\) make sense?)
Does it matter where you put your numbers? For example, if you decide to place a \(4\), would it make a difference if you put it in the middle or in a corner?
Are there any numbers that are good to use more than once?
Explorations
Play the game a few times, encouraging students to fill out their cards differently based on observations they're making. In order to guide their reasoning, you may want to intersperse some of the following explorations between games.
Monte Carlo simulation: Many students will notice that certain numbers come up more often than others. You can try to find these experimentally by making a lot of rolls and keeping track of the data. If you split students into pairs and have one roll and the other record, they can keep a tally of the number of times each sum is rolled. Combining this into a classroom set with 100+ data points should nicely illustrate the distribution. Below is an example with 200 data points:
| Sum | |
Frequency |
|---|---|---|
| \(2\) | |
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| \(3\) | |
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| \(4\) | |
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| \(5\) | |
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| \(6\) | |
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| \(7\) | |
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| \(8\) | |
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| \(9\) | |
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| \(10\) | |
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| \(11\) | |
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| \(12\) | |
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Combinatorial reasoning: Some students come into this activity with a sense that \(7\) is the most common value, often culturally-informed rather than mathematically. Asking "What are the ways that one can roll a \(7\)?" is often a good lead-in to an analytic exploration. For this, it's beneficial if the two dice are distinguishable -- one red die and one blue die, for example, allows students to more readily see the difference between \(2\)\(+\)\(5\) \(= 7\) and \(5\)\(+\)\(2\) \(= 7\). The most powerful tool for students here is making a table of all outcomes, but it's best if they come up with this idea themselves:
| \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | |
|---|---|---|---|---|---|---|
| \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
| \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
| \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) |
| \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) |
| \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
Wrap-Up Questions
If one number is more common than the others, is it a good idea to fill all nine spots on your card with that number?
Are there any anomalies in your data? (For example, \(3\) appears to be more common than \(4\) in our tally.) Can you explain this?
Is there a best way to fill out the card? What's your best card and why?
Extensions
Replacing addition with another operation can be a great source of other distributions. Instead of adding the two dice and calling out their sum \( a \) \( + \) \( b \), you can call out:
The difference \( a \) \( - \) \( b \) (i.e., subtract smaller from larger)
The maximum \( \max( \)\( a \) \( , \) \( b \)\( ) \)
The minimum \( \min( \)\( a \) \( , \) \( b \)\( ) \)
The product \( a \) \( \cdot \) \( b \)
The greatest common divisor \( \gcd( \)\( a \) \( , \) \( b \)\( ) \)
Students also like coming up with their own functions. The ones listed above are commutative, but with some care, the operation doesn't have to be.
If you have access to different dice, the following create interesting distributions with all sums \(12\) or less:
The sum of a \(4\)-sided die and an \(8\)-sided die
The sum of three \(4\)-sided dice
The sum of a coin (heads is \(1\), tails is \(2\)) and a \(10\)-sided die
The sum of six coins
To wring more victories out of each card, there can be a first winner for three-in-a-row, a second winner for two three-in-a-rows on the same card, and a third winner for blackout (all nine crossed out). If this version is played, the middle, corner, and edge squares all have different properties! How might this impact which numbers to choose and where to place them? The grid can also be enlarged to \(4 \times 4\) or bigger to explore these spatial questions.
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