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 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 $\frac{1}{2}$ 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

Experimental probability:

P_{\text{exp}}(H) = \frac{13}{20} = 0.65

Full solution

2
Theoretical probability: $P_{\text{theo}}(H) = 0.5$ (fair coin assumption)
3
Difference: $0.65 - 0.50 = 0.15$ — experimental exceeds theoretical by 15%
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

400

shots, a basketball player makes

260

. Estimate the probability she makes the next shot.

Example 4

medium

A spinner is supposed to be fair (3 equal colors). After

300

spins, red appears

80

times, blue

130

, green

90

. Is there evidence the spinner is unfair?

Example 5

hard

Compare: in

10

flips you get

7

heads (exp

P = 0.7

); in

10{,}000

flips you get

5050

heads (exp

P = 0.505

). Which estimate is more reliable for a fair coin and why?

Example 6

challenge

Why does flipping a coin

10

times sometimes give

7

heads and other times

3

? 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

0.5

. After 4 flips you got 3 heads. Does this make

P(\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

\frac{3}{200}

. Estimate the expected number of defects in a batch of 1000.

Example 13

medium

A spinner should land on blue

\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

\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

0.2

. You expect about how many wins in 50 plays, and would 8 wins be surprising?

Example 19

challenge

A die is rolled

n

times and shows a 6 exactly

\frac{n}{6}

times for

n=600

. As

n

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

0.25

. 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

4

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

Example 25

easy

You roll a die

30

times and get

7

fives. What is the experimental probability of rolling a five?

Example 26

medium

A die rolled

600

times produced

90

rolls of a

1

. Compare to the theoretical probability.

Example 27

medium

A weighted coin shows

60

heads in

100

flips. What is the experimental probability of heads?

Example 28

medium

A bag has

2

red and

3

blue marbles. What is the theoretical probability of red?

Example 29

medium

1000

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

Example 30

medium

You toss a thumbtack

50

times and it lands point-up

32

times. What is the experimental probability of point-up?

Example 31

hard

If theoretical

P = 0.4

and you perform

250

trials, what is the expected number of successes?

Example 32

hard

A simulation of

10{,}000

dart throws hits the bullseye

400

times. Estimate

P(\text{bullseye})

Example 33

hard

A die rolled

60

times shows

\{1: 8, 2: 12, 3: 9, 4: 11, 5: 10, 6: 10\}

. What is

P_{\text{exp}}(\text{even})

Example 34

hard

In a survey of

500

households,

325

have a dog. Estimate the probability a random household has a dog.

Example 35

hard

Theoretical

P = 0.25

. After

40

trials, you observe

14

successes. Is the experimental probability higher or lower?

Example 36

hard

A coin is flipped

1000

times, heads

480

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

0.02

per day. Over

365

days, the observed failure count is

12

. Is this consistent with the model?

Example 39

challenge

You bet on rolling a

6

on a fair die. After

60

rolls you see

20

sixes. What's the experimental probability vs. theoretical, and what should you do next?

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Related Concepts

Probability Sample Space Law of Large Numbers (Intuition)Sampling Distribution Hypothesis Testing

Background Knowledge

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

probabilitysample space