Chance experiments
🎯 In this topic you will
- Carry out and analyse experiments involving chance
- Examine experiments with both large and small sample sizes
- Compare relative frequencies with theoretical probabilities
🧠 Key Words
- relative frequency
Show Definitions
- relative frequency: The proportion of times an outcome occurs in a set of trials, calculated as $\tfrac{\text{number of successes}}{\text{total trials}}$, and used as an estimate of probability.
🎲 Relative Frequency of Rolling a Six
Zara rolls a dice 50 times. She is looking for sixes. Here are the results:
2 1 6 5 3 3 1 6 4 4
6 1 3 3 5 6 5 5 3 6
6 3 4 4 6 1 4 1 2 2
3 2 3 6 6 6 5 3 5 3
3 5 5 4 5 6 3 1 5 1
The top row shows the first ten rolls. The frequency of a 6 in the top row is 2. The relative frequency of a 6 after the first ten rolls is $\tfrac{2}{10}=0.2$. After 20 rolls, the frequency of a 6 is 5 and the relative frequency is $\tfrac{5}{20}=0.25$.
This table shows the changing relative frequency:
| Rolls | Frequency | Relative frequency |
|---|---|---|
| 10 | 2 | 0.2 |
| 20 | 5 | 0.25 |
| 30 | 7 | 0.233 |
| 40 | 10 | 0.25 |
| 50 | 11 | 0.22 |
You can show these values on a graph:
The theoretical probability of getting a 6 when you roll a dice is $\tfrac{1}{6}=0.167$ to 3 d.p. The relative frequency will keep changing as Zara rolls the dice more times.
❓ EXERCISES
1. Two coins are flipped together $25$ times. Both coins land on tails $3$ times.
a. Work out the relative frequency of two tails.
The experiment is repeated. This time both coins land on tails $7$ times.
b. Work out the relative frequency of two tails for the second experiment.
c. Put the two sets of results together and work out the relative frequency of two tails.
👀 Show answer
a. $P=\tfrac{3}{25}=0.12$
b. $P=\tfrac{7}{25}=0.28$
c. Total tails $=3+7=10$, total trials $=25+25=50$. $P=\tfrac{10}{50}=0.2$
2. Here is a spinner. The spinner is spun $200$ times.
This table gives the frequencies of each colour:
| Colour | Frequency |
|---|---|
| red | $78$ |
| white | $54$ |
| blue | $68$ |
a. Work out the relative frequency of each colour. Give your answers as decimals.
b. Each colour is equally likely. Compare the relative frequencies with the probability of each colour.
👀 Show answer
a. Total $=200$. $P(\text{red})=\tfrac{78}{200}=0.39$, $P(\text{white})=\tfrac{54}{200}=0.27$, $P(\text{blue})=\tfrac{68}{200}=0.34$
b. If equally likely, each colour $= \tfrac{1}{3}\approx0.333$. Results are close but not exact due to randomness.
3. Marcus rolls a dice $100$ times. He counts the total number of sixes after each set of $10$ rolls. Here are his results:
| Rolls | Total frequency | Relative frequency |
|---|---|---|
| $10$ | $2$ | $0.2$ |
| $20$ | $4$ | $0.2$ |
| $30$ | $5$ | $0.167$ |
| $40$ | $8$ | ? |
| $50$ | $9$ | ? |
| $60$ | $10$ | ? |
| $70$ | $11$ | ? |
| $80$ | $16$ | ? |
| $90$ | $17$ | ? |
| $100$ | $18$ | $0.18$ |
a. Copy and complete the table. Round the relative frequencies to $3$ decimal places if necessary.
b. Draw a graph to show how the relative frequency changes.
c. Draw a horizontal line to show the probability of a $6$.
👀 Show answer
a. $40: \tfrac{8}{40}=0.200$, $50: \tfrac{9}{50}=0.180$, $60: \tfrac{10}{60}=0.167$, $70: \tfrac{11}{70}=0.157$, $80: \tfrac{16}{80}=0.200$, $90: \tfrac{17}{90}=0.189$
b. Graph should show relative frequency fluctuating around $0.167$.
c. Draw line at $y=0.167$.
4. Sofia flips a coin $100$ times. She records the frequency of heads every $20$ flips. Here are the results:
| Flips | Frequency of heads | Relative frequency |
|---|---|---|
| $20$ | $8$ | $0.4$ |
| $40$ | $19$ | ? |
| $60$ | $30$ | ? |
| $80$ | $38$ | ? |
| $100$ | $44$ | ? |
a. Calculate the missing relative frequencies. Copy and complete the table.
b. Draw a graph to show the changing relative frequencies.
c. Compare the relative frequencies with the probability of the coin landing on a head.
👀 Show answer
a. $40: \tfrac{19}{40}=0.475$, $60: \tfrac{30}{60}=0.500$, $80: \tfrac{38}{80}=0.475$, $100: \tfrac{44}{100}=0.440$
b. Graph shows values fluctuating around $0.5$.
c. Theoretical probability $=0.5$. Relative frequencies are close to $0.5$ but vary due to randomness.
❓ EXERCISES
5. You can answer this question with a partner. You will need two dice.
a. Roll two dice. Record whether the total is $7$ or more. Repeat this $50$ times. After each $10$ rolls, work out the relative frequency of $7$ or more. Record your results in a table as shown.
| Rolls of two dice | $10$ | $20$ | $30$ | $40$ | $50$ |
|---|---|---|---|---|---|
| Frequency of $7$ or more | |||||
| Relative frequency |
b. Show your relative frequencies on a graph.
c. Use your graph to estimate the probability of a total of $7$ or more.
d. Compare your results with another pair. Do you have similar graphs? Do you have the same estimate for the probability?
👀 Show answer
a–b. Results will vary. Plot the relative frequency after each block of $10$ rolls.
c. Theoretical estimate: $P(\text{sum}\ge 7)=\tfrac{21}{36}=\tfrac{7}{12}\approx 0.583$.
d. Different pairs should get graphs fluctuating around $\tfrac{7}{12}$; small differences are due to randomness and sample size.
6. There are $20$ black and white balls in a bag. Arun takes out one ball at random and records the colour. Then he replaces the ball in the bag. He repeats this $200$ times. After every $20$ balls, he counts the frequency of a black ball. Here are his results:
| Draws | $20$ | $40$ | $60$ | $80$ | $100$ | $120$ | $140$ | $160$ | $180$ | $200$ |
|---|---|---|---|---|---|---|---|---|---|---|
| Frequency | $10$ | $14$ | $27$ | $36$ | $42$ | $50$ | $55$ | $62$ | $70$ | $79$ |
| Relative frequency | $0.5$ | $0.350$ | $0.450$ | $0.450$ | $0.420$ | $0.417$ | $0.393$ | $0.388$ | $0.389$ | $0.395$ |
a. Calculate the relative frequencies. Copy and complete the table.
b. Show the relative frequencies on a graph.
c. Estimate the numbers of black and white balls in the bag.
👀 Show answer
a. Values shown in the table: for example, $\tfrac{14}{40}=0.350$, $\tfrac{27}{60}=0.450$, … , $\tfrac{79}{200}=0.395$ (to $3$ d.p.).
b. Plot the points against draws; the graph should settle near a constant level.
c. The relative frequency approaches about $0.40$. With $20$ balls total, estimate $0.40\times 20=8$ black and $12$ white.
❓ EXERCISES
7. A calculator generates random digits between $0$ and $9$.
Arun generates $100$ digits. After each $20$ digits he counts the number of $0$s.
| Digits | $20$ | $40$ | $60$ | $80$ | $100$ |
|---|---|---|---|---|---|
| Frequency of $0$ | $2$ | $5$ | $7$ | $7$ | $8$ |
| Relative frequency |
a. Calculate the relative frequencies. Copy and complete the table.
b. Show Arun’s relative frequencies on a graph.
Marcus carries out the same experiment. Here are his results:
| Digits | $20$ | $40$ | $60$ | $80$ | $100$ |
|---|---|---|---|---|---|
| Frequency of $0$ | $2$ | $6$ | $8$ | $9$ | $15$ |
| Relative frequency |
c. Calculate the relative frequencies for Marcus. Copy and complete the table.
d. Show Marcus’ expected frequencies on the same graph as Arun’s.
Sofia generates $500$ digits. She finds the frequency of $0$ after every $100$ digits. Here are Sofia’s results:
| Digits | $100$ | $200$ | $300$ | $400$ | $500$ |
|---|---|---|---|---|---|
| Frequency of $0$ | $11$ | $27$ | $40$ | $52$ | $60$ |
| Relative frequency |
e. Work out Sofia’s relative frequencies. Copy and complete the table.
f. Show Sofia’s relative frequencies on a graph.
g. What is the probability that a digit is $0$? Compare this probability with the relative frequencies in the three experiments.
👀 Show answer
a. Arun’s relative frequencies: $\tfrac{2}{20}=0.100$, $\tfrac{5}{40}=0.125$, $\tfrac{7}{60}\approx 0.117$, $\tfrac{7}{80}=0.088$, $\tfrac{8}{100}=0.080$.
b. Plot these five values against digits $20,40,60,80,100$.
c. Marcus’ relative frequencies: $\tfrac{2}{20}=0.100$, $\tfrac{6}{40}=0.150$, $\tfrac{8}{60}\approx 0.133$, $\tfrac{9}{80}=0.113$, $\tfrac{15}{100}=0.150$.
d. On Arun’s graph, add Marcus’ expected/observed relative frequencies above. Compare the two traces.
e. Sofia’s relative frequencies: $\tfrac{11}{100}=0.110$, $\tfrac{27}{200}=0.135$, $\tfrac{40}{300}\approx 0.133$, $\tfrac{52}{400}=0.130$, $\tfrac{60}{500}=0.120$.
f. Plot these five points against digits $100,200,300,400,500$.
g. The theoretical probability a random digit is $0$ is $\tfrac{1}{10}=0.1$. Arun and Marcus’ values fluctuate around $0.1$ (small samples). Sofia’s larger sample still varies but tends toward a value near $0.1$; differences are due to randomness.
🧠 Think like a Mathematician
Context: A spreadsheet generated $500$ random digits between $0$ and $9$. The theoretical probability that a digit has any particular value is $0.1$.
Question: How closely do sample relative frequencies$0.1$ as the sample changes?
Method (work solo):
- Choose one digit (e.g., $7$) and a sample size (e.g., $n=50$). Select any $n$ entries from the grid (row by row or using a systematic rule).
- Count how many times your chosen digit appears: call this $f$.
- Compute the relative frequency: $\text{RF}=\dfrac{f}{n}$.
- Repeat with a different sample of the same size and the same digit; compare your two values of $\text{RF}$.
- Optional extension: choose several digits (e.g., any of $\{0,1,2\}$) and treat “success” as “in the set”. Then the theoretical probability is $\dfrac{k}{10}$ where $k$ is the number of digits in your set.
Follow-up tasks (from the sheet):
👀 Show Answer
- a. Example choice: digit $7$, sample size $n=50$. Suppose it appears $f=4$ times. Relative frequency $\text{RF}=\dfrac{4}{50}=0.08$.
- b. New independent sample of size $50$. If $f=7$, then $\text{RF}=0.14$. (Your values will vary.)
- c. Both estimates (e.g., $0.08$ and $0.14$) differ from the theoretical $0.1$ due to randomness. With larger samples (e.g., $n=200$ or all $500$ digits), the relative frequency typically gets closer to $0.1$ (law of large numbers).
- d. Example design: “success” = digit in $\{0,1,2\}$ so theoretical probability $\dfrac{3}{10}=0.3$. If in a sample of $n=100$ the count is $f=28$, then $\text{RF}=0.28$, reasonably close to $0.3$. Discrepancies are expected from sampling variation.