Experimental vs Theoretical Probability

Grade 4

Two ways to find probability

Theoretical probability is what you calculate. You assume all outcomes are equally likely and use the formula. A fair coin: P(heads) = ½. You never need to actually flip it.

Experimental probability (also called relative frequency) is what you measure. You run an experiment — flip the coin, spin the spinner, roll the dice — count the results, and estimate from real data.

Experimental probability = number of times event occurred ÷ total number of trials

Worked example

EXAMPLE

A spinner is spun 200 times. It lands on blue 70 times. (a) What is the experimental probability of blue? (b) If the spinner were fair with 4 equal sections, what would the theoretical probability of blue be?

(a) Experimental probability = 70 ÷ 200 = 0.35. (b) Theoretical probability = 1 ÷ 4 = 0.25. The experimental value is higher than expected — this suggests the spinner might be biased towards blue.

More trials = better estimate

The more trials you run, the closer experimental probability gets to the true theoretical value. With only 10 flips of a fair coin you might get 7 heads — that looks odd. With 10 000 flips you'll be very close to 5000 heads.

Exam questions sometimes ask: "How could you make the estimate more reliable?" The answer is always increase the number of trials.

When to use each

Use theoretical probability when a situation is fair and you can list all outcomes. Use experimental probability when you're dealing with something biased, real-world, or unknown — like the probability a drawing pin lands point-up. You can't calculate that; you have to measure it.

Try this one

A factory tests \( 1500 \) light bulbs. \( 36 \) of them are found to be faulty.

(a) Work out the relative frequency of a bulb being faulty. Give your answer as a decimal. [2]

(b) The factory produces \( 20\,000 \) bulbs in a week. Estimate how many will be faulty. [2]

Nova's hint:

This uses relative frequency as an estimate of probability, then applies it to a larger batch.

Ask yourself: 'what fraction of the tested bulbs were faulty, and how do I scale that up?' Once you have the relative frequency, multiply it by the larger total.

Try to Solve it with Nova?