D. Data · Data Literacy and Probability

Experiments with Two Independent Events

D2.2 D2.1 A1About 50 minutes

Learning goals

I can run an experiment with two independent events and find the experimental probability.
I can compare experimental and theoretical probability and explain why they can be different.
I can explain why more trials usually give results closer to the theoretical probability.

Words to know

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

trial
One try of an experiment, like flipping two coins once.

expected number
Theoretical probability × number of trials. It's the best prediction, not a promise.

fair
Every outcome is equally likely, like a normal coin or die.

Remember from Grade 5

Theoretical probability is what should happen: favourable outcomes ÷ total outcomes. Experimental probability is what did happen: number of times ÷ number of trials. In Grade 5 you compared them for one event. Now you'll test two independent events together.

The two-coin experiment

In the last lesson you found that two coins have 4 equally likely outcomes: HH, HT, TH, TT. Each has a theoretical probability of 1/4 = 25%.

So in 40 trials we expect each outcome about 25% × 40 = 10 times.

Owen flipped two coins 40 times. Here are his results.

Owen's results: HH 12, HT 9, TH 8, TT 11. The dashed line shows the expected number, 10 of each.

Outcome	Times	Experimental probability	Theoretical probability
HH	12	12/40 = 0.30 = 30%	1/4 = 25%
HT	9	9/40 = 0.225 = 22.5%	25%
TH	8	8/40 = 0.20 = 20%	25%
TT	11	11/40 = 0.275 = 27.5%	25%
Total	40	100%	100%

Owen's 40 trials compared with the theoretical probabilities.

Why aren't they the same?

Owen got HH 12 times, not 10. Did he do something wrong? No! Chance outcomes vary. In a small number of trials, results often wander away from what we expect.

Then Owen's whole family flipped two coins until they had 400 trials altogether:

HH: 104 → 104/400 = 26%
HT: 101 → about 25% (25.25%)
TH: 97 → about 24% (24.25%)
TT: 98 → about 25% (24.5%)

With more trials, the experimental probabilities got much closer to 25%. This usually happens: the more trials, the closer experimental probability tends to get to theoretical probability.

With 40 trials the bars wander from 20% to 30%. With 400 trials every bar lands close to the dashed 25% line.

Worked example

Example 1: One of each

Use Owen's 40 trials. Find the experimental probability of getting one head and one tail. Compare it with the theoretical probability.

"One of each" is HT or TH.
Experimental: 9 + 8 = 17 times. 17/40 = 0.425 = 42.5%.
Theoretical: 2 of the 4 outcomes, so 2/4 = 50%.
Owen's result is 7.5 percentage points lower than expected. With only 40 trials, that's not surprising.
In the 400 family trials: 101 + 97 = 198. 198/400 = 49.5%, very close to 50%.

Experimental 42.5% (40 trials) vs theoretical 50%. With 400 trials, 49.5%, much closer.

Worked example

Example 2: Coin and die

Lily flips a coin and rolls a die 60 times. She gets heads and 6 a total of 7 times. Compare with the theoretical probability.

Theoretical: H6 is 1 of 12 outcomes. P = 1/12 ≈ 0.083 ≈ 8.3%.
Expected number: 1/12 of 60 = 60 ÷ 12 = 5 times.
Experimental: 7/60 ≈ 0.117 ≈ 11.7%.
7 is a little more than 5. With only 60 trials, this is normal variation. To check whether her coin and die are fair, she would need many more trials.

Experimental about 11.7%, theoretical about 8.3%. The difference is small for only 60 trials.

Stay patient, stay curious

Probability experiments take patience. Flipping coins 40 times can feel long, and the results may surprise you. A surprising result isn't a mistake. It's a chance to ask "Why?" Working with others to pool data makes the job faster and the results more trustworthy. That's teamwork!

Just for fun

True story: more than 100 years ago, the scientist Karl Pearson flipped a coin 24 000 times. He got 12 012 heads, or about 50.05%. That's a lot of flipping for a number we already knew was about 50%!

Hands-on activity

Two-coin challenge

You need: two coins (or one coin flipped twice), paper and a pencil. A partner is helpful.

Make a tally chart with four rows: HH, HT, TH, TT. (Use two different coins, like a nickel and a dime, so you can tell HT from TH: the nickel is always first.)
Predict how many of each you'll get in 40 trials. (About 10 each.)
Flip both coins 40 times and tally the results.
Find each experimental probability as a fraction, decimal and percent.
Compare with the theoretical probability of 25%. Which outcome was closest? Which was farthest?
Combine your results with a family member's 40 trials to make 80 (or more). Did the percents get closer to 25%?

Outcome	H1	H2	H3	T1	T2	T3	Total
Times	22	18	21	19	23	17	120

Use this table for the Your turn questions: a coin flipped and the 1-2-3 spinner spun 120 times.

Your turn

What is the theoretical probability of each outcome (like H1)?

How many times would you expect each outcome in 120 trials?

What is the experimental probability of T2? Write it as a percent.

Which outcome came up the least? How far was it from the expected number?

Find the experimental probability of heads (with any number).

Do these results suggest the coin or spinner is unfair? Explain.

If the experiment were done 1200 times, about how many times would you expect H1? Would you expect exactly that?

More practice: warm-up

Experimental probability = ______ ÷ ______.

Two coins, 40 trials. How many times do you expect TT?

Owen got HH 12 times in 40 trials. What is the experimental probability of HH as a percent?

True or false: if experimental probability is different from theoretical probability, the experiment was done wrong.

What usually happens to experimental probability as the number of trials grows?

More practice: on your own

Two dice are rolled 36 times. How many times would you expect a sum of 7?

In 36 rolls, Ava got a sum of 7 9 times. Find her experimental probability and compare.

Owen's 40 trials: find the experimental probability of at least one head. Compare with the theoretical probability.

Why is it a good idea to pool (combine) results from many people?

Coin and die, 120 trials. How many times would you expect tails and an odd number?

Challenge

A spinner is supposed to have 4 equal sections. Spun with a coin flipped 800 times, the outcome "section A and heads" came up 210 times. Is something strange going on? Explain.

Owen's 40 trials had HH 30% and the 400 trials had HH 26%. Explain, using the idea of trials, why the 400-trial result is more trustworthy.

Design an experiment to test whether rolling a die and flipping a coin gives "heads and an even number" about 25% of the time. Say how many trials, how you'd record, and what result would make you think something is unfair.

Check your understanding

Quiz

Two coins are flipped 80 times. About how many times do you expect HT?

In 50 trials, an event happened 15 times. What is the experimental probability?

Which usually gives an experimental probability closer to the theoretical probability?

Owen got HH 30% of the time in 40 trials. The theoretical probability is 25%. What is the best conclusion?

Coin and die, 60 trials. What is the expected number of H6?

Two coins, 400 trials, "one of each" happened 198 times. What is the experimental probability?

For parents and teachers

What this lesson teaches: running experiments with two independent events, finding experimental probabilities, and comparing them with theoretical probabilities (D2.2, D2.1). Being patient with long experiments, staying curious about surprising results and pooling data with others are Strand A skills (A1).

How to help at home:

Do the two-coin experiment together and combine your results.
When results are "off", ask "Is that a big difference for this many trials?"
Use two different coins so HT and TH can be told apart.

Common difficulties:

Thinking experimental results should match theory exactly.
Deciding something is unfair after only a few trials.
Forgetting to divide by the number of trials (not the number of outcomes).
Treating HT and TH as the same outcome.