10 or 10,000 rolls of a die
If you roll a die 100 times, how many times will you get a 6?
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Relative frequency of each face
The height of each bar is the face's relative frequency: how many times it came up divided by the total number of rolls.
Relative frequency of 6 as the rolls add up
Face 6 Other faces
Show the numbers behind the charts
Each face
| Face | Times | Relative frequency |
|---|
The 6 as the rolls add up
| After | Sixes | Frequency of 6 |
|---|
What came right after three 6s in a row
Face 6 Other faces
Show the numbers behind the chart
| Next face | Times | Relative frequency |
|---|
How many times does 6 come up?
Choose how many rolls to make, lock in your guess for the 6, and tap Roll.
10 vs. 10,000
Do at least one round with 10 rolls and one with 10,000. Then compare the range of the 6 faces in Your rounds, below Roll.
You still need a round with 10 and one with 10,000.
Three 6s in a row
You rolled a 6 three times in a row. On the next roll, is the chance of a 6 less than, equal to, or greater than ?
If you haven't read it yet, read What's going on first, right below.
The test rolls the die 1,000,000 times, finds every time three 6s came up in a row, and records the roll that came next.
Make your guess first. The test unlocks when you tap Lock in my guess or I don't know.
What about 1,000?
You saw the range of the 6 faces with 10 and with 10,000. Before you run 1,000 rolls, predict: what will the highest relative frequency among the 6 faces be?
Now do a round of 1,000 down below.
Make your guess first. Roll unlocks when you tap Lock in my guess or I don't know.
Your rounds
Every round you finish shows up here.
| Rolls | Range of the 6 faces | Sixes | Frequency of 6 | Guess |
|---|
Range of the 6 faces: the lowest and the highest relative frequency among the six faces in that round.
What's going on
On a fair die, all 6 faces have the same chance of coming up. The probability of each one is , the dashed line in the charts.
The rest of the explanation unlocks when you do task 2, "10 vs. 10,000".
With 10 rolls, the relative frequency can end up far from : one face comes up 3 times, another not at all. The more rolls, the closer the relative frequency of each face gets to . This is called the law of large numbers. Look at the "Range of the 6 faces" column in Your rounds: it gets narrower as the number of rolls goes up.
The law is about relative frequency, not the count. In 10,000 rolls, the 6 almost never comes up exactly 1,667 times (10,000 ÷ 6 ≈ 1,667). It might come up 1,640 or 1,700 times. A difference of about 30 looks big, but divided by 10,000 it comes to 0.3 percentage points: the frequency of 6 stays between 16.4% and 17.0%. The die doesn't make up for anything. The new rolls only dilute the old ones.
For teachers
Objective
By the end, students can explain that the more times a fair die is rolled, the closer the relative frequency of each face gets to . It is the bridge between theoretical probability (counting cases) and empirical probability (observing frequencies), which the book introduces in section 7.5.
Prerequisites
Fractions, division with decimals, and percentages. Reading bar charts and line graphs. Students don't need to know anything about probability beyond "a 1 in 6 chance".
How to use in class
- Project the activity and read the opening question (2 min). Have each student write a guess for 100 rolls in their notebook. Ask a few students to share theirs out loud.
- Do a round of 10 and a round of 100 with the class (3 min). Ask: did anyone get it right? Was a guess of 17 "the right one"?
- Let pairs work on their own for 5 minutes on tasks 1 and 2 in the panel. The goal is to do at least one round of each size and look at the Your rounds table.
- Bring the class back together (5 min) and read "What's going on" aloud. Ask students to explain in their own words what the range of the 6 faces shows.
- Before the challenge (task 3), take a show of hands: less, equal, or greater (2 min). Then lock in the class's guess, run the test on the projector, and discuss (5 min).
Common mistakes
Thinking that with many rolls the 6 comes up exactly N ÷ 6 times. It shows up as a guess of 1,667 out of 10,000 and as frustration when the result is 1,650. The round feedback shows the difference in the count and in the relative frequency: the count was off by 17, but the frequency landed 0.2 percentage points from 16.7%.
Gambler's fallacy. A student who picks "less" thinks the 6 "has already come up too much"; one who picks "greater" thinks the die "is on a hot streak". Both imagine that the die remembers the past. Ask: how would the die know what came up before?
Thinking the die "makes up for it" to get back to . A student who says "after a lot of 6s there have to be fewer 6s" has this misconception. The last sentence of "What's going on" (diluting, not making up) helps.
Discussion questions
- Why can a face fail to come up even once in 10 rolls?
- If the count of 6s drifts away from N ÷ 6, how can the relative frequency be getting closer to ?
- A lottery draws numbers that "haven't come up in months." Do they have a better chance now?
- A team won 47 of 80 games. Can you say its chance of winning the next one is about 59%? What could make that estimate a bad one?
Based on Contemporary Mathematics, OpenStax, section 7.5, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.