Inquiry Question The game Dungeons and Dragons requires ...

Math 8-9, Unit 7

Inquiry Question The game Dungeons and Dragons requires rollingdice to activate different skills, can I calculate and understand the probability of rolling dice while fighting a dragon?

Name: ___________________________ Date: ____________

General Instructions You are playing a game of Dungeons and Dragons with some friends, and you want to understand the probability and math behind the rolling of the dice. Dungeons and Dragons (DND) is a role-playing game where you roll various sided dice to try and hit an enemy, do damage, and more. The whole game is based off rolling dice and probability. You will be learning about the probability of rolling dice, how to calculate probabilities of multiple dice rolls,and how to visualize probability.

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Math 8-9, Unit 7

Materials you’ll need:

Pen or pencil Calculator Dice (if you have some) Computer (or large tablet, phone is probably too small)

Project submission:

Submit the completed pages of this project.

Review

Before getting into the game, let’s review how simple probability works. There are 3 types of probabilities you will need to know how to calculate. Let’s look at the definition of each of them and an example or two each.

1. Single event

a. Probability is a type of ratio where we compare how many times something could happen to all the possible outcomes.

Probability=Thenumber of wanted outcomesThenumber of possibleoutcomes

Example: If you want to know the probability of rolling a 5 on a six-sided dice (which we will call a d6), then we would write

P=wanted outcomespossibleoutcomes

=16=16.7

There was 1 wanted outcome (getting a 5), and 6 possible outcomes (1,2,3,4,5,6). Thus, the probability of getting a 5 is 16.7%.

2. Independent events

a. Two events are independent if the outcome of the first event doesn’t affect the secondevent. To calculate the probability of two independent events, we multiply the probability of the first event by the probability of the second event.

P ( X∧Y )=P ( X )∗P(Y )

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Math 8-9, Unit 7

Example: If you had 3 6-sided dice (3d6), what is the probabilityof rolling three 4s?

P (4∧4∧4 )=P ( 4 )∗P (4 )∗P (4 )=

16∗1

6∗1

6=

1216

=0.5

3. Mutually exclusive events

a. Two events are mutually exclusive if the events cannot happen at the same time. Instead of thinking “what’s the probability of getting a 4 and another 4”, we would think “what is the probability of getting a 4 or 5”. Probability is then written as:

P ( X∨Y )=P ( X )+P (Y )

Example: Say you roll 1d6, and you want to know the probability of rolling a 1 or 2.

P (1∨2 )=P (1 )+P (2 )=16+

16=

26=

13=33

So, the probability of rolling a 1 or 2 on a single d6 is 33%. But what about if there aremultiple dice?

Example 2: You now have 2d6’s, and you want to know the probability of rolling atotal of 3, in other words, you want the numbers on the two d6’s to add to 3. To dothis, we need to think of all the ways that we can get a total of 3 between 2 dice. Our calculation will look like

P (1∧2∨2∧1 ) ,

since we could have dice #1 be a 1 and dice #2 be a 2, or the other way around, dice #1 be a 2 and dice #2 be a 1.

P (1∧2∨2∧1 )=P (1∧2 )+P (2∧1 )=P (1 )∗P (2 )+P (2 )∗P(1)

We used the rules of independent and mutually exclusive events to rewrite our probability. The “or” turned into a “+”, and the “and” turned into a “*”. Now,

P (1 )∗P (2 )+P (2 )∗P (1 )=

16∗1

6+

16∗1

6=

136

+136

=236

=118

=5.6

So, the probability of rolling 2d6’s and getting a total of 3 is 5.6%. Woo!

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Math 8-9, Unit 7

Time to play some Dungeons and Dragons

Story

The story begins with you and your party of adventurers exploring a cave system. You are a ranger elf, and you specialize in your bow and arrow. Your group has heard rumors of an ancient dragon thatlives deep in the cave and has a hoard of riches that it has stolen from burning down towns. You told your party that you’re here for the riches, but really what you want is to teach the dragon a lesson so that it never destroys another village again. As you are crossing a bridge, you feel a gust of wind blowup from the depths, followed by a deep gurgling rumble.

It’s here.

Suddenly the large creature flies up past your party, and you are filled with dread as you see it’s immense size. The ancient dragon sends a cone of fire into the air, and your party readies themselves for battle.

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Math 8-9, Unit 7

Probability

You loose an arrow at the dragon, and you need to roll a 20-sided dice (1d20) to see if your shot hits. Your roll must be higher than 12 to hit the dragon.

Please give all your probabilities as percent’s. What is the probability of getting any number (1,8,13,19, etc.) when rolling 1d20?

What is the probability that you hit (get higher than 12)?

You rolled a 16 and hit the dragon! Now you need to roll for how much damage to the dragon you inflict. You need to roll 2d6’s (2 six-sided dice).

What is the probability that you do the maximum amount of damage (12)?

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Math 8-9, Unit 7

You rolled 2d6’s and got a 9. What is the probability that you rolled a 9?

Hint: how many possible ways can you get a 9 rolling 2d6’s?

Digging deeper

You did 9 damage to the dragon! Let’s think about this a little. You calculated the probability of rolling a 12 and a 9 with 2d6’s. Why do you think rolling a 9 is more probable than 12?

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To explore this idea, we will be using the website https://anydice.com/ to observe how probability is affected by the number of dies rolled and the number of sides on the dice. Any Dice allows you to pick any number of dies with any number of sides and graph what the probability of rolling each number is. It allows you to visually see the most and least probable outcome. Let’s walk through how the website works first.

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https://anydice.com/

Math 8-9, Unit 7

This is the full page of the website. You want to make sure you’ve clicked on “Calculator” on the lop left.

This is where you tell the website the type and number ofdies you want to calculate probabilities for. The code for this is very simple, and has 4 parts:

1. Output tells the website to calculate/display something. This will always be the first word and won’t ever change.

2. This is where you tell the website the numberof dies (the 1) and the number of sides on the die (the 6).

3. This command tells the website to link the information about the dice (in this picture the 1d6) to what you want it to be called.

4. The part inside the “ “ is the name for the dice rolled. I made the name the same as the type of dice I’m rolling to keep it straightforward.

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Math 8-9, Unit 7

You can add multipledice by hitting “enter” onthe keyboard and writinga new line.

This part is where you see your data and decide how you want it displayed.

On the top left you can see the “calculate” button, which you press when you’ve typed some code and want to see the graphs.

The “View” tabs allow you to select how you want your data displayed.

The table at the bottom shows your data. The “#” column is the number on the dice, the “%” column is the probability that the # is rolled, and the table shows you how big those probabilities are.

By clicking the “graph” button in the “View” tabs, you can see a graph of the probabilities of rolling numbers. In the graph above there is a plot for 1d6 and 1d20. We calculated the probability of rolling any number on a 1d20 earlier to be 5%, and we see that in the graph!

Plotting Probabilities

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Math 8-9, Unit 7

Now that we can navigate the website, lets return to our question frombefore…

Why do you think rolling a 9 is more probable than 12?

In AnyDice, type in this code: output 2d6 named “2d6”. This adds 2d6’s onto the program. Click “calculate” and click “graph”. On the next page, draw the graph that you see and add the percentages and dice numbers.

Based off the graph above, explain why rolling a 9 is more probable than rolling a 12. What is the most common number to roll, and why?

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Math 8-9, Unit 7

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This is called a dice chart, and it shows the probability of rolling 2d6’s, just like the graph above. It visually shows you how many different combinations of dice are needed to get a total of some number. For example, to get a 3 by rolling 2d6’s, you can either roll a 1 on a dice and 2 on the other, or you can roll a 2 on the first dice, and a 1 on the other. It’s a great way to visualize probability!

Back to the DRAGON

The dragon had taken a hit from you, and that didn’t make it happy. It launches itself into the air and begins to fly away…wait…it’s coming back! You run for cover behind a large rock, and just intime! A torrent of hungry fire wraps around the boulder, and although it misses you, the heat is almost unbearable.

Your party and you fight the dragon for a while longer, and it appears to be getting weary. You think that maybe with one more good hit, the dragon will not be able to fight anymore, and maybe then it will stop destroying towns and villages.

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Math 8-9, Unit 7

You ready a spell called “Ice Spike” and launch it at the dragon! You roll 1d20to see if you hit, and you need to get 16,17,18,19, or 20 to hit! What is theprobability that you hit (get higher than 15)?

You rolled a 20! That’s a critical hit, and you do double damage! You realize your “Ice Spike” spell gives you two choices of which dice to roll. You can either roll 2d3’s or 3d2’s.

The big question is, which option will give me a greater probability of rolling a 5 or 6?

You will explore this question using dice charts, calculations, and confirming your results with AnyDice!

Dice Charts

To get a visual picture of the probabilities of your two options 2d3’s or 3d2’s, you will fill in 2 dice charts with all the possible options.

Fill in the dice chart with the values needed to equal the total # on the left. Look at the dice chart on the previous page for guidance if needed.

Calculate the probability of getting a 5 or 6 with 2d3’s using the methods in the “Review” section.

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Math 8-9, Unit 7

Now that we know the probability of rolling a total of 5 or 6 using 2d3’s, let’s repeat the above process again for 3d2’s.

Fill in the dice chart with the values needed to equal the total # on the left and fill in the probability column on the right.

Calculate the probability of getting a 5 or 6 with 3d2’s using the methods in the “Review” section.

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Math 8-9, Unit 7

Confirm your results by plotting both 2d3 and 3d2 on AnyDice, then copy the graph onto the grid below.

Finally, in words, explain which option will give you a better chance of rolling a 5 or 6 and why.

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Math 8-9, Unit 7

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The conclusion of the story

You rolled a 6 and because you had a critical hit you do DOUBLE DAMAGE. You whip an icicle as large as tall man over your shoulder and towards the dragon, and it shatters against its chest,spraying shards of ice all throughout the cave. The dragon recoils and begins to retreat.

Your party goes on to search for the riches, but you return to town, content in the fact that the dragon will no longer destroy villages. That night, the nearby villages celebrate you and your party, and a sense of peace and security returns to the people that knew nothing but fear for many years.

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Math 8-9, Unit 7

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Inquiry Question The game Dungeons and Dragons requires ...

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