Probability Experiments

Probability is a number that tells you how likely something is to happen. Flip a fair coin and you expect heads about half the time; roll a fair die and you expect a 4 about one time in six. When you actually run the trials and count what you saw, the fraction (favorable outcomes) / (total trials) is the experimental probability — the answer the experiment gave you.

The Probability Formula

• P(event) = favorable outcomes / total trials.
• Favorable outcomes = the number of times the event you care about happened.
• Total trials = the number of times you ran the experiment in total.
• Every probability is a number from 0 to 1, inclusive.

Reading the 0-to-1 Scale

• P = 0 means the event is impossible — it cannot happen.
• P = 1/4 (or 0.25) means the event is unlikely — it happens about one time in four.
• P = 1/2 (or 0.5) means an even chance — it happens about half the time.
• P = 3/4 (or 0.75) means the event is likely — it happens about three times in four.
• P = 1 means the event is certain — it happens every single time.

Worked Examples

1. A coin is flipped 20 times and lands on heads 13 times. P(heads) = 13/20 = 0.65.
2. A standard die is rolled 60 times and shows a 4 on 8 rolls. P(4) = 8/60 = 2/15.
3. A spinner is spun 25 times and stops on red 15 times. P(red) = 15/25 = 3/5 = 0.6.
4. A bag of 10 marbles has 0 green marbles. P(green) = 0/10 = 0 (impossible).
5. The same bag has 10 yellow marbles. P(yellow) = 10/10 = 1 (certain).

Experimental vs Theoretical

Theoretical probability is what you expect from the structure of the experiment — like 1/2 for heads on a fair coin. Experimental probability is what you actually measured. With only a few trials they can disagree a lot; with many trials, they usually drift close together. Running more trials is the simplest way to improve an experimental estimate.