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=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.

Theoretical probability is what SHOULD happen in a perfect world: a fair coin should land heads 50%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%60\%). The more times you flip, the closer your experimental result gets to 50%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 is computed from equally-likely outcomes; experimental probability is the observed fraction from real trials, and they converge as trials grow.

Common stuck point: The procedure for experimental vs. theoretical probability is the easy part; the trap is reporting 12\frac{1}{2} when the problem gives trial results. Asking "Does the probability come from reasoning about the possible outcomes (theoretical) or from counting what occurred in real trials (experimental)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the probability come from reasoning about the possible outcomes (theoretical) or from counting what occurred in real trials (experimental)?

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.

Answer

Experimental: 0.65. Theoretical: 0.50. Differ by 0.15; converge with more flips (LLN).

First step

1
Experimental probability: Pexp(H)=1320=0.65P_{\text{exp}}(H) = \frac{13}{20} = 0.65

Full solution

  1. 2
    Theoretical probability: Ptheo(H)=0.5P_{\text{theo}}(H) = 0.5 (fair coin assumption)
  2. 3
    Difference: 0.65−0.50=0.150.65 - 0.50 = 0.15 — experimental exceeds theoretical by 15%
  3. 4
    Convergence: by the Law of Large Numbers, as more flips are conducted, experimental probability converges to theoretical (0.5)
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.

Example 3

medium
In 400400 shots, a basketball player makes 260260. Estimate the probability she makes the next shot.

Example 4

medium
A spinner is supposed to be fair (3 equal colors). After 300300 spins, red appears 8080 times, blue 130130, green 9090. Is there evidence the spinner is unfair?

Example 5

hard
Compare: in 1010 flips you get 77 heads (exp P=0.7P = 0.7); in 10,00010{,}000 flips you get 50505050 heads (exp P=0.505P = 0.505). Which estimate is more reliable for a fair coin and why?

Example 6

challenge
Why does flipping a coin 1010 times sometimes give 77 heads and other times 33? Is the coin unfair?

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.

Example 3

easy
A fair six-sided die is rolled. What is the theoretical probability of rolling a 4?

Example 4

easy
A coin was flipped 50 times and landed heads 28 times. What is the experimental probability of heads?

Example 5

easy
A spinner has 5 equal sections. What is the theoretical probability of landing on a given section?

Example 6

easy
In 200 trials, an event occurred 60 times. What is its experimental probability?

Example 7

easy
As the number of trials increases, experimental probability tends to do what relative to theoretical probability?

Example 8

easy
A bag has 3 red and 7 blue marbles. What is the theoretical probability of drawing red?

Example 9

easy
A die rolled 10 times gave the number 6 zero times. A student concludes the die can never roll a 6. Is this valid?

Example 10

easy
Theoretical probability of heads is 0.50.5. After 4 flips you got 3 heads. Does this make P(heads)=0.75P(\text{heads})=0.75?

Example 11

medium
A die is rolled 60 times; a 3 appears 8 times. Compare the experimental and theoretical probabilities of rolling a 3.

Example 12

medium
A factory's experimental defect rate is 3200\frac{3}{200}. Estimate the expected number of defects in a batch of 1000.

Example 13

medium
A spinner should land on blue 14\frac14 of the time. In 400 spins it landed blue 95 times. Compute the experimental probability and compare.

Example 14

medium
Two events: drawing a king from a deck (theoretical) vs. a survey estimating the chance a random adult owns a car (experimental). Which event's probability must be estimated from data?

Example 15

medium
A coin is flipped 1000 times, landing heads 530 times. What experimental probability does this give, and is it strong evidence of bias?

Example 16

medium
A standard die's theoretical probability of an even number is 12\frac12. In 30 rolls, 18 were even. Find the experimental probability and its difference from theoretical.

Example 17

medium
To estimate the probability that a thumbtack lands point-up, why must we use experimental rather than theoretical probability?

Example 18

medium
A game has theoretical win probability 0.20.2. You expect about how many wins in 50 plays, and would 8 wins be surprising?

Example 19

challenge
A die is rolled nn times and shows a 6 exactly n6\frac{n}{6} times for n=600n=600. As nn grows from 6 to 600 to 6000, what happens to the GAP between experimental and theoretical probability, and why?

Example 20

challenge
Two students each flip a coin: one flips 10 times (6 heads), the other 1000 times (530 heads). Whose experimental probability is more trustworthy as an estimate of the true value, and why?

Example 21

challenge
A medical test's theoretical false-positive rate is unknown, so a lab runs it on 5000 healthy patients and gets 100 positives. Estimate the false-positive probability and explain why theoretical reasoning could not give it.

Example 22

medium
A spinner's theoretical probability of green is 0.250.25. Over 1200 spins, about how many greens do you expect?

Example 23

easy
What is the theoretical probability of flipping heads on a fair coin?

Example 24

easy
A spinner has 44 equal sections labeled A, B, C, D. What is the theoretical probability of landing on C?

Example 25

easy
You roll a die 3030 times and get 77 fives. What is the experimental probability of rolling a five?

Example 26

medium
A die rolled 600600 times produced 9090 rolls of a 11. Compare to the theoretical probability.

Example 27

medium
A weighted coin shows 6060 heads in 100100 flips. What is the experimental probability of heads?

Example 28

medium
A bag has 22 red and 33 blue marbles. What is the theoretical probability of red?

Example 29

medium
In 10001000 trials of a fair coin, you expect about how many heads?

Example 30

medium
You toss a thumbtack 5050 times and it lands point-up 3232 times. What is the experimental probability of point-up?

Example 31

hard
If theoretical P=0.4P = 0.4 and you perform 250250 trials, what is the expected number of successes?

Example 32

hard
A simulation of 10,00010{,}000 dart throws hits the bullseye 400400 times. Estimate P(bullseye)P(\text{bullseye}).

Example 33

hard
A die rolled 6060 times shows {1:8,2:12,3:9,4:11,5:10,6:10}\{1: 8, 2: 12, 3: 9, 4: 11, 5: 10, 6: 10\}. What is Pexp(even)P_{\text{exp}}(\text{even})?

Example 34

hard
In a survey of 500500 households, 325325 have a dog. Estimate the probability a random household has a dog.

Example 35

hard
Theoretical P=0.25P = 0.25. After 4040 trials, you observe 1414 successes. Is the experimental probability higher or lower?

Example 36

hard
A coin is flipped 10001000 times, heads 480480 times. Is the experimental probability close to fair?

Example 37

hard
A standard deck. Compute the theoretical probability of drawing a heart from a single shuffle.

Example 38

challenge
A factory's machines fail with theoretical probability 0.020.02 per day. Over 365365 days, the observed failure count is 1212. Is this consistent with the model?

Example 39

challenge
You bet on rolling a 66 on a fair die. After 6060 rolls you see 2020 sixes. What's the experimental probability vs. theoretical, and what should you do next?
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Background Knowledge

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

probabilitysample space