Experimental vs. Theoretical Probability Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Experimental vs. Theoretical Probability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Theoretical probability is calculated from known outcomes (P = \frac{\text{favorable}}{\text{total}}), while experimental probability is estimated from actual trials (P \approx \frac{\text{times event occurred}}{\text{total trials}}). As the number of trials increases, experimental probability tends to approach theoretical probability.

Theoretical probability is what SHOULD happen in a perfect world: a fair coin should land heads 50\% of the time. Experimental probability is what ACTUALLY happens when you try it: flip a coin 20 times and you might get heads 12 times (60\%). The more times you flip, the closer your experimental result gets to 50\%—that's the law of large numbers in action.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Theoretical probability uses logic and counting. Experimental probability uses data. They converge as the number of trials grows large (law of large numbers).

Common stuck point: A small number of trials can give very misleading results. Getting 4 heads in 5 flips doesn't mean P(\text{heads}) = 80\%—you need many more trials.

Worked Examples

Example 1

easy
A coin is flipped 20 times: 13 heads. Compare experimental probability of heads to theoretical probability. Explain why they differ and when they converge.

Solution

  1. 1
    Experimental probability: P_{\text{exp}}(H) = \frac{13}{20} = 0.65
  2. 2
    Theoretical probability: P_{\text{theo}}(H) = 0.5 (fair coin assumption)
  3. 3
    Difference: 0.65 - 0.50 = 0.15 — experimental exceeds theoretical by 15%
  4. 4
    Convergence: by the Law of Large Numbers, as more flips are conducted, experimental probability converges to theoretical (0.5)

Answer

Experimental: 0.65. Theoretical: 0.50. Differ by 0.15; converge with more flips (LLN).
Experimental probability is calculated from observed outcomes; theoretical probability is derived from mathematical models. Small samples produce large discrepancies; large samples converge (LLN). Neither is 'wrong' — they measure different things.

Example 2

medium
A thumbtack is tossed 200 times: 130 times it lands point-up. Calculate the experimental probability. Explain why we must use experimental (not theoretical) probability here.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A die is rolled 60 times. Theoretical expected count for each face: 10. Actual counts: 1→8, 2→11, 3→9, 4→12, 5→10, 6→10. Calculate experimental probability for rolling a 1 and compare to theoretical.

Example 2

hard
A simulation model predicts 25% of customers churn per month. After 6 months of actual data: 28%, 22%, 26%, 24%, 27%, 23%. Calculate the experimental mean, compare to theoretical, and determine if the model is reasonable.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probabilitysample space