D. Data · Data Literacy and Probability

Experiments: Comparing Experimental and Theoretical Probability

D2.2 D2.1 A1About 55 minutes

Learning goals

I can carry out an experiment with two independent events and with two dependent events.
I can find the experimental probability from my results.
I can compare experimental and theoretical probabilities and explain why they may differ.
I can explain why more trials usually give results closer to the theoretical probability.

Words to know

trial
One run of an experiment, like drawing two cards once.

experimental probability
Number of times the event happened ÷ number of trials.

theoretical probability
Number of favourable outcomes ÷ number of possible outcomes, worked out by reasoning.

simulation
An experiment that models a real situation, like using cards to model choosing names.

Remember from Grade 6

You compared experimental and theoretical probabilities for two independent events and saw that more trials usually brought the results closer together. Now you'll test dependent events too, and see a result that surprises many people.

A surprising question

Take 4 playing cards: 2 red and 2 black. Shuffle them face down. Draw 2 cards without putting the first one back.

Which is more likely: the two cards are the same colour, or different colours?

Many people guess "50-50". Let's work it out, then test it.

Worked example

Example 1: The theoretical probability

Name the cards R1, R2, K1, K2 (K for black). Two cards are drawn without replacement. Find P(same colour).

Total outcomes: 4 × 3 = 12.
Same colour: R1R2, R2R1, K1K2, K2K1. That is 4 outcomes.
P(same colour) = 4/12 = 1/3 ≈ 33.3%. So P(different colours) = 8/12 = 2/3.

1/3. Different colours are twice as likely as the same colour! After the first card, only 1 card of 3 matches its colour.

After the first card is drawn and kept out, only one of the three cards left is the same colour. That is why P(same colour) is 1/3, not 1/2.

For comparison, with replacement the events are independent: 4 × 4 = 16 outcomes, and same colour has 2 × 2 + 2 × 2 = 8 of them. P(same colour) = 8/16 = 1/2. Not putting the card back makes "same colour" less likely.

Experimental probability

Experimental probability = number of times the event happened ÷ number of trials

An experiment won't match the theory exactly, because of chance. With a small number of trials, the results can be far off. With many trials, the experimental probability usually gets closer to the theoretical probability.

Who	Trials	Same colour	Experimental probability
Ava (first try)	10	5	5/10 = 50%
Ava	30	9	9/30 = 30%
Ava's group	100	36	36/100 = 36%
Whole class	300	102	102/300 = 34%

Sample results for drawing 2 of 4 cards without replacement. Theoretical probability: 1/3 ≈ 33.3%.

Experimental percentages for 10, 30, 100 and 300 trials: 50%, 30%, 36%, 34%. The dashed line shows the theoretical probability, 1/3 ≈ 33.3%. With more trials, the bars get closer to the line.

Worked example

Example 2: Compare experimental and theoretical

Ava did 10 trials and got the same colour 5 times. She says, "See? It's 50-50!" Use the class data to respond.

Ava's experimental probability: 5/10 = 50%. Theoretical: 33.3%.
10 trials is a small number, so chance can push the result far from the theory.
The whole class did 300 trials: 102/300 = 34%, very close to 33.3%.

With more trials, the results support the theory: same colour happens about 1 time in 3, not 1 time in 2.

Worked example

Example 3: Predicting from probability

(a) Two coins are flipped 80 times. About how many times would you expect two heads? (b) The 4-card experiment is done 60 times. About how many times would you expect the same colour?

(a) Coins are independent. P(two heads) = 1/4. Expected: 1/4 × 80 = 20 times.
(b) P(same colour) without replacement = 1/3. Expected: 1/3 × 60 = 20 times.
These are predictions. Real results will usually be close, but not exact.

(a) about 20 times, (b) about 20 times.

When results are far off

If an experiment with many trials is still far from the theory, look for a reason: Were the cards shuffled well? Did someone peek? Was a card put back by mistake? Is the coin or spinner fair? Comparing results is how we find mistakes in the theory or in the experiment. Recording carefully and sharing data with others makes everyone's results stronger.

Just for fun

Flip a coin 10 times and get 7 heads? The coin isn't magic. Flip it 1 000 times and the share of heads will usually land close to 50%. Patience is an experimenter's superpower!

Hands-on activity

The two-colour card experiment

You need: 4 playing cards (2 red and 2 black), paper, a pencil and a partner if possible.

Predict: in 30 trials, how many times will you get the same colour? (Theory says about 10.)
Without replacement: shuffle the 4 cards face down, draw 2, record same or different, then return both cards and shuffle again. Do 30 trials.
Find your experimental probability and compare it with 1/3.
With replacement: draw 1 card, record it, put it back, shuffle, draw again. Do 30 trials. Compare with 1/2.
Combine your results with a family member's or a friend's results. Does the larger set get closer to the theory?

Math talk: "Why is 'same colour' less likely when you don't put the first card back?"

Your turn

In 50 trials of the 4-card experiment (without replacement), Leo got the same colour 18 times. What is his experimental probability as a fraction and a percent?

Compare Leo's 36% with the theoretical probability.

Two coins are flipped 40 times. About how many times would you expect one head and one tail?

Jada did 36 trials of rolling a die twice. She got a double (both rolls the same) 9 times. Theoretical P(double) = 6/36 = 1/6. Compare.

Why do more trials usually give an experimental probability closer to the theoretical one?

With replacement, P(same colour) with 2 red and 2 black cards is 1/2. Without replacement, it is 1/3. Explain the difference.

More practice: warm-up

A coin is flipped 20 times and lands heads 12 times. Experimental P(heads)?

What is the theoretical P(heads) for a fair coin?

What does "trial" mean?

P(event) = 1/4. About how many times in 100 trials?

Which is more reliable: 10 trials or 1 000 trials?

More practice: on your own

A bag has 3 green and 2 blue marbles. Two are drawn without replacement, 50 times. About how many times would you expect both blue?

In that experiment, Sara got both blue 4 times in 50 trials. Experimental probability? Is that reasonable?

A spinner has 4 equal sections. It is spun twice, 64 times. Theoretical P(same section both times) = 1/4. Predict the number of times.

In 200 trials of the 4-card experiment, a group got the same colour 98 times (49%). The theory says 1/3. Give two possible reasons.

Design a simulation with cards to model this: "Two students are chosen from 3 girls and 2 boys to read the morning announcements." What would you do?

Challenge

With 3 red and 3 black cards, two drawn without replacement: find P(same colour). Is it closer to 1/2 than with 2 and 2? Why?

Mo did 20 trials and got 50%. Pia did 200 trials and got 36%. The theoretical probability is 33.3%. Whose result should you trust more, and what is the combined experimental probability?

A game: draw 2 of the 4 cards (2 red, 2 black) without replacement. You win 1 point if they are the same colour, and your friend wins 1 point if they are different. Is the game fair? How could you make it fair?

Check your understanding

Quiz

A spinner is spun 40 times and lands on blue 12 times. What is the experimental P(blue)?

Two cards are drawn without replacement from 2 red and 2 black. What is the theoretical P(same colour)?

Which is most likely to be close to the theoretical probability?

P(two heads) when flipping two coins is 1/4. About how many times in 200 flips of two coins?

A group's experimental probability is far from the theory after 500 trials. What should they do?

Why is P(same colour) higher with replacement than without?

For parents and teachers

Big idea: D2.2 asks students to determine and compare theoretical and experimental probabilities of two independent and two dependent events. The 4-card experiment is a good one: most people expect "same colour" to be 50-50, but without replacement it is only 1/3. This challenges a preconceived notion with both reasoning and data.

Common difficulties:

Expecting experiments to match theory exactly. Small samples vary a lot.
Forgetting to put both cards back between trials (that changes the experiment).
Mixing up "with" and "without" replacement.

How to help: Do the card experiment together, 30 trials each, and combine your results. Celebrate careful recording and honest results, even when they don't match the theory. This builds A1 skills: perseverance, teamwork and critical thinking.