L1-03 · Repeat and record
A probability is what a model predicts. A frequency is what you actually count. With more trials, the frequency usually gets closer to the probability, but it rarely lands on it exactly.
Level 1 · Grades 4–6 · about 40 minutes
Big idea. A probability is what a model predicts. A frequency is what you actually count. With more trials, the frequency usually gets closer to the probability, but it rarely lands on it exactly.
Before this lesson. L1-02, L0-05
Grown-ups. The guide for adults is at the end of this page.
What you will learn¶
I can find the relative frequency of an event from my own counts.
I can explain the difference between a probability and a frequency.
I can explain why a few trials are not enough to judge a coin or a die.
Story: Does this coin like heads?¶
Leo flips a coin 10 times and gets 7 heads. “This coin likes heads!” he says. Maya shakes her head. “Our model says a fair coin lands heads with probability one half. Ten flips is not much. Let’s keep going and write down every result.”
After 20 flips, they have 11 heads. That is 11 out of 20, a little more than half. After 50 flips, they have 25 heads, which is 25 out of 50, exactly one half for a moment.
They keep flipping until they reach 200 flips. They count 101 heads. The relative frequency is 101 out of 200. That is just a tiny bit more than one half.
“So it was not a heads-loving coin,” says Leo. “The 7 out of 10 was just a wobbly start.” Maya nods. “The probability stayed one half the whole time. Our counts got closer to it as we did more trials.”
The fraction of flips that were heads, after each of 200 flips.
What the animation shows. A line shows the fraction of flips that were heads so far. It starts high, at 7 out of 10. It wobbles, then stays close to a dashed line at one half. Dots mark 11 out of 20, 25 out of 50, and, at the end, 101 out of 200.
Hands-on: Coin flip record¶
About 20 minutes.
You need:
A coin
A paper cup to shake it in (optional)
Graph paper or a printed grid
A pencil and a ruler
A calculator (optional)
Steps:
Make a table with three columns: flips so far, heads so far, and fraction of heads.
Flip the coin 10 times. Count the heads. Write the totals in the first row.
Work out the fraction of heads. For example, 6 heads in 10 flips is 6 out of 10, or 0.6.
Flip 10 more times. Add to your totals, so the second row shows 20 flips. Work out the new fraction.
Keep going until you reach 50 flips, one row for every 10.
On graph paper, draw a dashed line across at one half. Plot your five fractions and join the dots.
Look at your graph. Did the dots get closer to the dashed line as you flipped more?
Practice together¶
After 20 flips you have 9 heads. What is the relative frequency of heads? Does that prove the coin is unfair?
Check your answer
It is 9 out of 20, which is 0.45. That is close to one half. Twenty flips are far too few to call a coin unfair.
Try it yourself¶
1. A spinner has 4 equal parts, and 1 part is blue. What is the probability of blue? You spin 40 times and get blue 13 times. What is the relative frequency of blue?
Check your answer
The probability of blue is one fourth, or 0.25. The relative frequency is 13 out of 40, which is 0.325. They are different, and that is normal for only 40 spins.
2. Sam rolls a fair die 6 times and never gets a 6. Sam says the die must be broken. What would you tell Sam?
Check your answer
Getting no 6 in 6 rolls happens about one time in three, so it is not a sign that the die is broken. The probability of a 6 is one sixth on every roll, but that does not promise one 6 in every 6 rolls. Sam should roll many more times before deciding.
Talk about it¶
Why were 10 flips not enough to judge Leo’s coin?
Does a coin remember what it landed on before?
When would you trust counts more than a model? When would you trust the model more?
Check yourself¶
1. A fair coin is flipped 200 times and lands heads 101 times. Which statement is true?
A. The coin is unfair because 101 is not 100.
B. The relative frequency of heads is 101 out of 200.
C. The probability of heads is 101 out of 200.
Show the answer
B. The relative frequency of heads is 101 out of 200.
101 out of 200 is what was counted, so it is a frequency. The probability for a fair coin stays one half. A small gap, like 101 instead of 100, is normal.
2. Which fraction of heads is usually closer to one half?
A. The fraction of heads in 10 flips
B. The fraction of heads in 1,000 flips
C. Both are always the same distance from one half
Show the answer
B. The fraction of heads in 1,000 flips
With more flips, the fraction of heads usually settles near the probability. With only a few flips, it can be far away.
3. Remember Lesson 2. A bag has 2 red counters and 6 blue counters. What is the probability of drawing red? (Review of L1-02)
A. 6 out of 8, which is three fourths
B. 2 out of 6, which is one third
C. 2 out of 8, which is one fourth
Show the answer
C. 2 out of 8, which is one fourth
There are 8 counters in all, and 2 of them are red. So the probability of red is 2 out of 8, which is one fourth.
Words to know¶
- probability
- A number from 0 to 1 that says how likely something is, according to a model.
- relative frequency
- How often something really happened, written as a fraction: times it happened divided by total tries.
- trial
- One try of an experiment, like one coin flip.
For grown-ups: about this lesson
Why it matters. The lesson separates a model’s probability from an observed relative frequency. That is the heart of the Common Core probability standards that come next, approximating a probability from data and comparing a model with observed frequencies (7.SP.6 and 7.SP.7) NGA Center and CCSSO, 2010. OpenStax describes probability as long-run relative frequency and states the law of large numbers with coin tosses Barbara Illowsky and Susan Dean, 2023. The animation uses the course’s tested simulation with a fixed random seed, so the numbers in the story, the video, and this page all match.
Watch for these mix-ups.
The fraction of heads must reach exactly one half in the end. (It gets close in the long run; it does not have to hit one half.)
After several heads, tails is due. (Each flip is separate; the coin has no memory.)
A small sample far from the model proves the model wrong. (Small samples vary a lot; judge with many trials.)
The gap between the heads count and the tails count always shrinks. (The share of heads gets closer to one half, but the count gap can grow.)
To make it easier. Do 30 flips in three rows of 10, and write fractions out of 10 only.
To make it harder. Pool the flips of the whole family or class (several hundred) and plot the running fraction. Then roll a die and track the fraction of 6s, which should settle near one sixth.
Limits. The lesson treats the coin as fair. Real coins and flips can be very slightly biased, and no number of flips can prove that a coin is perfectly fair.
On the computer. Grown-ups can see where this idea leads later in Lab: Chance and repeated trials, written for older learners.
Sources for this guide. NGA Center and CCSSO, 2010Barbara Illowsky and Susan Dean, 2023
Next: L1-04 · Count within a condition
- NGA Center and CCSSO. (2010). Common Core State Standards for Mathematics. Common Core State Standards Initiative. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
- Barbara Illowsky and Susan Dean. (2023). 3.1 Terminology. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-1-terminology