Theoretical vs. Experimental Probability

By the end of this lesson, you will be able to distinguish theoretical from experimental probability, calculate each, and explain the Law of Large Numbers.

Theoretical Probability

Theoretical probability is based on reasoning about equally likely outcomes — not on actually conducting experiments.

$$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$

This assumes all outcomes in the sample space are equally likely.

Example: Roll a fair 6-sided die. What is the theoretical probability of rolling a 4?

$P(4) = \frac{1}{6} \approx 0.167$

Example: What is the probability of picking a red card from a standard deck?

$P(\text{red}) = \frac{26}{52} = \frac{1}{2}$

Experimental Probability

Experimental probability is based on actually running trials and counting results.

$$P(\text{event}) = \frac{\text{number of times event occurred}}{\text{total number of trials}}$$

This reflects what actually happened, not what should happen.

Example: A coin is flipped 100 times; heads appears 43 times.

$P(\text{heads}) = \frac{43}{100} = 0.43$ (experimental)

Compare to theoretical: $0.5$ — the results aren't exactly equal.

The Law of Large Numbers

As the number of trials increases, the experimental probability gets closer and closer to the theoretical probability.

With 10 flips: might get 7 heads (70%) — far from 50%.

With 1,000 flips: likely get something between 480 and 520 heads (48–52%).

With 10,000 flips: even closer to 50%.

Implication: don't conclude a coin is unfair from just a few flips. You need many trials for experimental probability to be meaningful.

Comparing Theoretical and Experimental

	Theoretical	Experimental
Based on	Reasoning about outcomes	Actual trials
Formula	favorable/total