Course 2.2Lesson 5 of 5

Project: a dice laboratory

In this project we build a dice laboratory. The program rolls a die 600 times with a seed so that the result is the same each run, counts each face with a dictionary, compares each count with the expected 100, finds the mean, the mode and the biggest gap from the expected count, and draws a bar chart in text. We plan what a fair die should show, build the program in steps with the random and statistics modules, test it with different seeds and different numbers of rolls, and extend it with two dice and the sum of two dice.

about 25 min1 simulation2 practice questions

Lesson 5 of 5Show all lessons

Try the code and the simulations beside the lesson.

What you will be able to do

After this lesson you can

use random and statistics together in one program,
count outcomes with a dictionary, and show them as a table and a bar chart,
compare what you see with what you expect, and
see how the pattern of chance appears with many trials.

The goal

Is a die fair? A fair die gives each face with the same chance, 1 in 6. If you roll it 600 times, you expect about 100 of each face. But "about" is the interesting word. Will it be exactly 100? Close to it? How close? You can find out in seconds, by rolling a virtual die 600 times.

Step 1: what to expect

Before we run any code, think.

For a fair die, the mean of one roll is 3.5, the average of 1, 2, 3, 4, 5 and 6.
For 600 rolls, we expect each face about 100 times.
The counts will not be exactly 100. Chance makes them wobble. A gap of 10 or 15 is normal. A gap of 50 would be strange.
The mode (the most common face) will be a different face in different runs. It tells nothing.

Step 2: the program

Run it. Look at how it is built.

- random.seed(1) makes the whole experiment repeat exactly. Everyone who runs it sees the same numbers. Change the seed to 2, and you will see different numbers.

- rolls = [random.randint(1, 6) for roll in range(600)] makes the list of 600 rolls in one line. This is a list comprehension, a short way to build a list with a loop. You can read it as "a randint for each of 600 rolls".

- The counting pattern with a dictionary counts each face. The key is the face, and the value is how many times.

- f"{face:>4} {counts[face]:>6} ..." lines up the columns of the table. >4 means a width of 4 with the text at the right.

- "#" * (counts[face] // 5) makes a bar, with one # for every 5 rolls. The bars show at a glance that the faces are almost equal.

- statistics.mean(rolls) is the mean of the rolls. It should be close to 3.5.

- max(abs(c - 100) for c in counts.values()) finds the biggest gap between a count and the expected 100.

Step 3: test and explore

A laboratory is for experiments. Try these, and write down what you expect before each.

Experiment	What do you expect?
a different seed	different counts, still close to 100 each
60 rolls instead of 600	counts near 10, but the gaps are bigger in proportion
60 000 rolls	each count very close to 10 000 in proportion
a loaded die, with `random.choice([1, 2, 3, 4, 5, 6, 6, 6])`	face 6 comes about three times in eight

The last two rows show the law of large numbers: the more trials, the closer the share of each face gets to the expected 1 in 6. The count can still wobble by a hundred or two, but as a share of the total it becomes tiny. This is the idea behind every poll and every simulation.

The bars are what to expect. Your random counts will wobble around them, and wobble less, in proportion, the more you roll.

Step 4: extend it

Pick one or two, and do them without looking at a solution. Write down what you expect first.

Two dice. Roll two dice and add them. Which sum comes most often? (Expect 7. Why? Count the ways: only one way to make 2, but six ways to make 7.)
The loaded die. Compare the table of a fair die and a loaded die. How many rolls do you need to notice the difference?
The first six. How many rolls does it take to get the first 6? Repeat 1000 times and find the average.
A date for the experiment. Show the date of the experiment with datetime.
The spread. Use statistics.pstdev on the six counts. Is it small?

Common mistakes

Running without a seed and then not being able to repeat the result.
Counting with range(1, 6). A die has six faces, and randint(1, 6) is the right call.
Judging a die from 10 rolls. A small sample says nothing.
Forgetting that "close to 100" is not "exactly 100".
Putting the seed inside the loop, so every roll is the same.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

Warm-up. Open the lab Roll and count, if you have not done it, and the lab Describe the data.
Build. Open the lab Project: the dice laboratory. The checker runs your function with a seed and compares with the true counts.
Stretch. Open the lab A fair die? and compare its output with your laboratory.

Remember these

Think about what to expect before you run an experiment.
A seed makes the experiment repeat. Change the seed to see the wobble.
Count with a dictionary. Show the table and a bar chart.
More trials make the shares closer to the expected value.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

How many times do you expect each face in 600 rolls?

Why will the counts not all be exactly 100?

Which sum comes most often with two dice, and why?

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Problems (1)

Roll two dice 6000 times, with the seed 3.Medium · High School (9–10) · by StemPy Lab

Python Labs (1)

Project: the dice laboratoryPython Lab · by StemPy Lab

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