Theoretical vs Experimental Probability

There are TWO ways to talk about the probability of an event. Theoretical probability is what you EXPECT to happen, worked out from the structure of the experiment - like 1/2 for heads on a fair coin or 1/6 for rolling a 4 on a fair die. Experimental probability is what ACTUALLY happens when you run the experiment and count - like 28 heads in 50 flips, which is 28/50 = 0.56.

Two Formulas, Two Questions

• Theoretical: P = favorable outcomes / total possible outcomes. Computed by COUNTING (no experiment).
• Experimental: P = favorable outcomes / total trials. Computed by RUNNING the experiment and counting what you saw.
• The two values are usually close but not equal - especially with a small number of trials.
• With many trials, experimental probability drifts toward the theoretical value (this is called the Law of Large Numbers).

Worked Example - Coin Flip

1. Theoretical: a fair coin has 2 outcomes (heads, tails) and 1 favorable (heads). P(heads) = 1/2 = 0.5.
2. Experimental: flip the coin 50 times and count heads. Suppose you got 28 heads.
3. Experimental P(heads) = 28/50 = 0.56.
4. Compare: 0.56 is close to 0.5, but not exactly 0.5. Run more trials and the gap usually shrinks.

Worked Example - Rolling a Die

1. Theoretical: a fair die has 6 outcomes. P(rolling a 4) = 1/6 (about 0.167).
2. Experimental: roll the die 30 times and count fours. Suppose you got 6 fours.
3. Experimental P(4) = 6/30 = 0.20.
4. Compare: 0.20 is close to 0.167 but a little high. With 600 rolls instead of 30 the experimental value would usually land much closer to 1/6.

Why Do They Disagree?

Random variation. Even a fair coin sometimes lands on heads 7 times in 10 flips just by chance. Theoretical probability tells you the long-run average; experimental probability is one snapshot from a finite number of trials. The fewer trials you run, the more the snapshot can wobble away from the true value.