Experimental Probability and Simulations

Sometimes you can’t work out a probability from a list of equally likely outcomes. What’s the chance a thumbtack lands point up? Or that a basketball player sinks a free throw? You find out by experimenting: doing many trials and seeing how often it happens. The more trials you do, the more you can trust the answer.

Key ideas

Experimental probability

$$P(A) \approx \frac{\text{number of times } A \text{ happened}}{\text{number of trials}}$$

This fraction is also called the relative frequency of $A$.

The law of large numbers

With only a few trials, experimental probability can be far from the theoretical value. As the number of trials grows, the experimental probability tends to get closer and closer to the theoretical probability.

The running proportion of heads in $500$ simulated coin flips.

In this simulation, the proportion of heads was $0.7$ after $10$ flips, $0.54$ after $100$, and $0.52$ after $500$.

Simulations

A simulation models an experiment using something easier to repeat, often random numbers. For example:

• a spreadsheet formula like =RANDBETWEEN(1,6) simulates rolling a die
• a random number from $0$ to $1$ that’s less than $0.3$ can stand for “it rains” when $P(\text{rain}) = 0.3$

Simulations are useful when an experiment is slow, expensive, or impossible to repeat many times.

Expected number of occurrences

If $P(A) = p$ and you do $n$ trials, you’d expect $A$ to happen about $np$ times.

Worked examples

Example 1: Comparing with theory

A coin is flipped $50$ times and lands heads $28$ times. Find the experimental probability of heads, and compare it with the theoretical probability.

Solution.

$$P(\text{heads}) \approx \frac{28}{50} = 0.56$$

The theoretical probability is $0.5$. A difference like this is normal with only $50$ trials.

Example 2: When there’s no theory

A thumbtack is dropped $120$ times and lands point up $78$ times. Estimate the probability of point up, and how many times it would land point up in $500$ drops.

Solution.

$$P(\text{point up}) \approx \frac{78}{120} = 0.65$$

In $500$ drops, expect about $0.65 \times 500 = 325$ point-up landings.

Example 3: Reading the law of large numbers

Using the graph above, explain why someone who flipped only $10$ coins might wrongly think the coin was unfair.

Solution. After $10$ flips, the simulation had $7$ heads, a proportion of $0.7$. That looks unfair, but with so few trials, big swings are common. As more flips were added, the proportion settled near $0.5$. Small samples can mislead; large ones are more reliable.

Example 4: Designing a simulation

Design a simulation to estimate the probability of rolling at least one $6$ in four rolls of a die.

Solution.

1. In a spreadsheet, put =RANDBETWEEN(1,6) in four columns of one row: that’s one trial of four rolls.
2. In a fifth column, check whether any of the four is a $6$.
3. Copy the row down to make many trials, say $1000$.
4. Divide the number of trials with at least one $6$ by $1000$.

The theoretical answer (from independent events) is $1 - \left(\tfrac{5}{6}\right)^4 \approx 0.518$, so a good simulation should give something close to that.

Common mistakes

Trusting a small number of trials. Ten trials can easily give $0.7$ for a fair coin. Use many trials before drawing conclusions.

Expecting the experimental value to match exactly. Even with many trials, experimental and theoretical probabilities rarely agree perfectly. They should just be close.

Thinking a coin “is due” for heads. Each flip is independent. The law of large numbers works because of many trials, not because the coin balances itself out.

Dividing by the wrong total. The denominator is the number of trials, not the number of outcomes.

Practice

1. (Warm-up) A player makes $26$ of $40$ free throws. What is the experimental probability of making a free throw?

Solution

$\tfrac{26}{40} = 0.65$

2. (Warm-up) A die is rolled $60$ times and shows a $6$ thirteen times. Compare the experimental and theoretical probabilities of rolling a $6$.

Solution

Experimental: $\tfrac{13}{60} \approx 0.217$. Theoretical: $\tfrac{1}{6} \approx 0.167$. The experimental value is a bit high, which is not surprising with only $60$ rolls.

3. (Warm-up) The probability of rain on a June day is $0.3$. About how many rainy days would you expect in June’s $30$ days?

Solution

$0.3 \times 30 = 9$ days.

4. (Core) A bag holds $20$ marbles in three colours. A marble is drawn, its colour recorded, and it’s put back, $100$ times: $48$ red, $31$ blue, $21$ green. Estimate how many marbles of each colour are in the bag.

Solution

Multiply each relative frequency by $20$: red $\approx 9.6$, blue $\approx 6.2$, green $\approx 4.2$. A good estimate is about $10$ red, $6$ blue, and $4$ green.

5. (Core) Two classes each flip a coin $30$ times. One gets $19$ heads and the other gets $13$. Is something wrong? How could they get a more reliable estimate?

Solution

Nothing is wrong: with only $30$ flips, results like $\tfrac{19}{30} \approx 0.63$ and $\tfrac{13}{30} \approx 0.43$ happen by chance. Combining the data ($32$ heads in $60$ flips, about $0.53$) or doing many more flips gives a more reliable estimate.

6. (Core) A cereal company puts one of $5$ different toys in each box, each equally likely. Design a simulation to estimate how many boxes you’d need to buy to collect all $5$.

Solution

Use =RANDBETWEEN(1,5) to represent opening a box. Keep generating numbers until all of $1$ to $5$ have appeared, and record how many boxes it took. Repeat many times (say $100$) and average the results.

(The theoretical average is about $11.4$ boxes, so a good simulation should land close to that.)

7. (Core) Two coins are flipped $1000$ times: two heads $260$ times, one head and one tail $497$ times, two tails $243$ times. Compare with the theoretical probabilities.

Solution

Theoretical: $P(\text{HH}) = 0.25$, $P(\text{one of each}) = 0.5$, $P(\text{TT}) = 0.25$.

Experimental: $0.260$, $0.497$, $0.243$. All very close, as expected with $1000$ trials.

8. (Challenge) A student rolls a die $60$ times and gets $25$ sixes. Should they suspect the die is unfair? What would you do to find out?

Solution

$\tfrac{25}{60} \approx 0.42$, far above $\tfrac{1}{6} \approx 0.17$. With a fair die, you’d expect about $10$ sixes in $60$ rolls, and $25$ is very unlikely by chance. That’s good reason to be suspicious. To be more confident, roll it many more times: if the proportion of sixes stays well above $0.17$, the die is probably unfair.

9. (Challenge) In Example 4, a class runs the simulation $200$ times and gets at least one $6$ in $109$ trials. How does this compare with the theoretical probability?

Solution

Experimental: $\tfrac{109}{200} = 0.545$. Theoretical: $1 - \left(\tfrac{5}{6}\right)^4 \approx 0.518$. They’re close; the difference of under $0.03$ is reasonable for $200$ trials.