Practice Experimental vs. Theoretical Probability in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

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?

Example 3

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 4

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

Example 5

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

Example 6

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

Example 7

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 8

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

Example 9

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 10

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

Example 11

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

Example 12

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 13

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

Example 14

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 15

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 16

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

Example 17

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

Example 18

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 19

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

Example 20

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