Grade by Grade
D. Data · Data and Probability

Theoretical and Experimental Probability

D2.2 D2.1 A1About 50 minutes
Learning goals
  • I can find the theoretical probability of an event.
  • I can do an experiment and find the experimental probability of an event.
  • I can compare theoretical and experimental probabilities.
  • I can explain why more trials usually bring the two closer together.

Words to know

theoretical probability
What should happen: favourable outcomes ÷ possible outcomes, when they are equally likely.
experimental probability
What did happen in an experiment: number of times the event happened ÷ number of trials.
trial
One try of an experiment, like one flip of a coin.
fair
A fair coin, die or spinner gives every outcome the same chance.

Is the coin broken?

Amir flipped a coin 20 times and got 12 heads. "This coin is unfair!" he said. "It should be 10 heads!"

Is he right? Let's find out what to expect, and what really happens.

Two kinds of probability

Theoretical probability is what should happen. You find it by thinking:

Theoretical P = favourable outcomes ÷ possible outcomes (when they are equally likely)

For a coin: P(heads) = 1/2.

Experimental probability is what did happen. You find it by doing an experiment:

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

For Amir: 12 heads in 20 flips. Experimental P(heads) = 12/20 = 3/5 = 0.6.

The theoretical probability is 1/2 = 0.5, and the experimental probability is 0.6. They are close but not the same. That's normal!

Worked example

Example 1: Compare for a die

Noor rolled a die 30 times. Her results: 1 → 4 times, 2 → 6, 3 → 5, 4 → 3, 5 → 7, 6 → 5. Compare the theoretical and experimental probability of rolling a 6, and of rolling an even number.

  1. Theoretical P(6) = 1/6. In 30 rolls, expect 1/6 of 30 = 5 sixes.
  2. Experimental P(6) = 5/30 = 1/6. Exactly the same this time!
  3. Theoretical P(even) = 3/6 = 1/2, so expect 15 of 30.
  4. Even rolls: 2 → 6, 4 → 3, 6 → 5. 6 + 3 + 5 = 14. Experimental P(even) = 14/30.
  5. 14/30 is very close to 15/30 = 1/2.

P(6): theoretical 1/6, experimental 5/30 = 1/6. P(even): theoretical 1/2 = 15/30, experimental 14/30. Very close.

Results of 30 Die Rolls012345678123456expected: 5 of eachNumber rolledNumber of timesSource: Noor's experiment
Noor's 30 rolls: 1 came up 4 times, 2 came up 6 times, 3 came up 5 times, 4 came up 3 times, 5 came up 7 times and 6 came up 5 times. The dashed line at 5 shows what we would expect for each number.

More trials, closer results

With only a few trials, results can be far from what we expect. With many trials, the experimental probability usually gets closer to the theoretical probability.

Amir's class put all their coin flips together: 100 flips and 48 heads. Experimental P(heads) = 48/100 = 0.48. That's much closer to 0.5 than Amir's 0.6!

So Amir's coin is probably fine. 12 heads in 20 flips is a normal result.

0.00.10.20.30.40.50.60.70.60Amir20 flips0.48Class100 flipstheoretical 0.5experimental P(heads)
Amir's 20 flips gave 0.6, which is 0.1 away from the theoretical 0.5. The class's 100 flips gave 0.48, only 0.02 away. More trials usually land closer to the theoretical probability.
Worked example

Example 2: Is the spinner fair?

A spinner has 4 equal sections: green, yellow, blue and white. Kai spins it 200 times and gets green 98 times. Is something strange going on?

  1. Theoretical P(green) = 1/4. Expect 1/4 of 200 = 50 greens.
  2. Experimental P(green) = 98/200 = 49/100 = 0.49.
  3. That's about twice as many as expected, after many trials.

This is very far from 1/4 after 200 spins. The spinner might not be fair (maybe it's bent or tilted). It's worth checking!

Stay curious, stay patient

Experiments take time, and the results can surprise you. A "weird" result isn't a mistake. It's data! Keep careful records, combine results with others to get more trials, and talk about what you notice.

Just for fun

Why didn't Amir's coin want to be part of the argument?

It didn't want to take sides: heads or tails!

Hands-on activity

Coin flip experiment

You need: a coin, paper and a pencil. (Optional: a die.)

  1. Predict: in 10 flips, how many heads? In 50 flips? (Theoretical: 5 and 25.)
  2. Flip the coin 10 times and tally heads and tails. Write the experimental P(heads) as a fraction.
  3. Flip 40 more times, for 50 in all. Write the new experimental P(heads) as a fraction out of 50, then out of 100 (multiply top and bottom by 2) and as a decimal.
  4. Compare: which result was closer to 1/2 = 0.5, after 10 flips or after 50?
  5. Bonus: roll a die 30 times and make a bar graph like Noor's. Draw a line at 5 for what you expect.

Your turn

1

What is the theoretical probability of rolling a 2 on a die?

2

Mia rolled a die 12 times and got a 2 three times. What is the experimental P(2)?

3

A coin is flipped 50 times and lands on heads 27 times. What is the experimental P(heads), as a fraction and a decimal?

4

What is the theoretical P(heads) as a decimal?

5

Noor's results: what is the experimental P(5)?

6

Will experimental and theoretical probability always be equal? Explain.

More practice: warm-up

1

Which kind of probability comes from doing an experiment?

2

Which kind comes from thinking about equally likely outcomes?

3

What is one try of an experiment called?

4

A spinner landed on blue 8 times in 10 spins. What is the experimental P(blue)?

5

In 40 coin flips, about how many heads do you expect?

More practice: on your own

1

A spinner has 5 equal sections, 2 of them green. Theoretical P(green)? In 50 spins, it landed on green 18 times. Experimental P(green)? Compare.

2

Noor's results: what is the experimental P(1 or 2)? What is the theoretical P(1 or 2)?

3

Jay flipped a coin 4 times and got 4 tails. He says the coin is unfair. Do you agree?

4

Which is more likely to give an experimental P(heads) close to 0.5: 10 flips or 1000 flips? Why?

5

A bag has 3 red and 1 blue cube. You pull one out without looking, record the colour, and put it back, 40 times. About how many reds do you expect?

Challenge

1

Four friends each flipped a coin 25 times. Heads: 15, 11, 13, 9. Find each friend's experimental P(heads) as a decimal, then combine all the flips.

2

A die was rolled 60 times and 6 came up 31 times. Compare with the theoretical probability. What might you conclude?

3

Design an experiment to test whether a paper cup is more likely to land on its side than on its top or bottom. What will you record, and how will you decide?

Check your understanding

Quiz

1.

What is the theoretical probability of flipping heads?

2.

A spinner landed on green 6 times in 20 spins. What is the experimental P(green)?

3.

What usually happens to the experimental probability as you do more trials?

4.

Amir got 12 heads in 20 flips. What's the best conclusion?

5.

In 60 die rolls, about how many times do you expect a 3?

6.

A spinner with 4 equal sections (1 green) landed on green 98 times in 200. What does this suggest?

For parents and teachers

What this lesson teaches: finding and comparing theoretical probability (what should happen) and experimental probability (what did happen in trials) (D2.2), linked to fractions on the probability line (D2.1). Patience, perseverance and curiosity when results surprise us (A1) are built in.

How to help at home:

  • Do the coin or die experiments together. Combining family results into more trials makes the big idea clear.
  • Use trial counts like 10, 20, 25, 50 or 100 so the fractions turn into decimals easily.

Common difficulties:

  • Expecting experiments to match theory exactly, especially with few trials.
  • Deciding a coin or die is unfair after only a handful of trials.
  • Writing experimental probability as "times it happened ÷ times it didn't" instead of ÷ total trials.
  • Thinking past results change the next one ("I got 4 tails, so heads is due"). Each flip of a fair coin is still 1/2.