Experimental vs. Theoretical Probability Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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: Pexp(H)=1320=0.65P_{\text{exp}}(H) = \frac{13}{20} = 0.65
  2. 2
    Theoretical probability: Ptheo(H)=0.5P_{\text{theo}}(H) = 0.5 (fair coin assumption)
  3. 3
    Difference: 0.65−0.50=0.150.65 - 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.

About Experimental vs. Theoretical Probability

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

Learn more about Experimental vs. Theoretical Probability →

More Experimental vs. Theoretical Probability Examples

Example 2 medium

A thumbtack is tossed 200 times: 130 times it lands point-up. Calculate the experimental probability

Example 3 easy

A die is rolled 60 times. Theoretical expected count for each face: 10. Actual counts: 1→8, 2→11, 3→

Example 4 hard

A simulation model predicts 25% of customers churn per month. After 6 months of actual data: 28%, 22

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