Experimental vs. Theoretical Probability Math Example 2

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

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.

Solution

  1. 1
    Experimental probability: P(point-up)=130200=0.65P(\text{point-up}) = \frac{130}{200} = 0.65
  2. 2
    Why experimental only: unlike a coin or die, a thumbtack has no symmetry argument for theoretical probability; the probability depends on its specific shape, weight distribution, and surface
  3. 3
    Theoretical probability requires a mathematical model based on symmetry or known distribution; for asymmetric objects, we must rely on empirical data
  4. 4
    Best estimate: P(point-up)≈0.65P(\text{point-up}) \approx 0.65 from the experiment

Answer

Pexp(point-up)=0.65P_{\text{exp}}(\text{point-up}) = 0.65. Must use experimental probability — no theoretical model exists for irregular objects.
Theoretical probability requires known, symmetric outcomes (dice, coins, cards). For real-world objects (thumbtacks, defect rates, weather), probability must be estimated experimentally from data. This is also the basis for frequentist statistics: probability = long-run relative frequency.

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 1 easy

A coin is flipped 20 times: 13 heads. Compare experimental probability of heads to theoretical proba

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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