Pre-Algebra
Probability (introductory)
Experimental vs. theoretical probability
20 practice questions
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Theory + worked examples
Experimental vs Theoretical Probability
California Pre-Algebra • Standard 7.SP.6 • Probability
Experimental vs Theoretical Probability is a topic in Probability in the California Common Core State Standards. It is aligned to Standard 7.SP.6, which requires students to compare experimental and theoretical probability.
Theoretical probability comes from equally likely outcomes; experimental probability comes from trials and approaches theory over time.
Theory
Theoretical probability is computed from equally likely outcomes; experimental probability comes from actual trials.
With more trials, experimental probability approaches the theoretical value.
Experimental approaches theoretical.
The two kinds.
Experimental probability:
\[P_{\text{exp}}=\dfrac{\text{successes}}{\text{trials}}\]
The law of large numbers links the two.
How to compare them
- Find theoretical from equally likely outcomes.
- Find experimental from actual results.
- Compare the two values.
- Expect closer agreement with more trials.
Example 1 — Theoretical
Theoretical \(P(\text{heads})\) on a fair coin?
Solution
One of two.
| \(P\) | \(=\) | \(\dfrac{1}{2}\) |
Example 2 — Experimental
\(23\) heads in \(50\) flips. Find the experimental probability.
Solution
Successes over trials.
| \(\dfrac{23}{50}\) | \(=\) | \(0.46\) |
Example 3 — Convergence
What happens with more trials?
Solution
Experimental probability approaches theoretical.
Example 4 — Predict
Expected heads in \(200\) flips (fair coin)?
Solution
Half of \(200\).
| \(0.5\cdot200\) | \(=\) | \(100\) |
Common pitfalls
Experimental can differ from theoretical in few trials.
More trials narrow the gap.
Theoretical needs equally likely outcomes.
Frequently asked questions
What is theoretical probability?
Computed from equally likely outcomes.
What is experimental probability?
Found from actual trials.
Do they always match?
Not exactly, but they get closer with more trials.
Experimental P of 23 heads in 50?
\(0.46\).
More in Probability (introductory)