Theoretical Probability Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Theoretical Probability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Theoretical probability is the expected probability of an event calculated by mathematical reasoning about equally likely outcomes, without conducting experiments. It equals the number of favorable outcomes divided by the total number of possible outcomes.

For a fair coin, you KNOW heads is 12\frac{1}{2} without flipping. You calculate based on logic: 1 favorable outcome (heads) out of 2 possible outcomes. That's theoretical - it's what SHOULD happen.

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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 starts by naming the possible outcomes and the event rule before assigning or combining probabilities.

Common stuck point: Students often know a procedure related to theoretical probability but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

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A bag holds 6 white, 4 black, and 2 red marbles. What is the theoretical probability of drawing a marble that is not white?

Answer

P=12P = \frac{1}{2}

First step

1
Total marbles: 6+4+2=126+4+2=12.

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Example 2

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A deck has 52 cards. What is the theoretical probability of drawing a red card or a queen?

Example 3

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A box has 5 red, 4 green, and 3 blue balls. Two balls are drawn without replacement. What is the theoretical probability that both are red?

Example 4

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Three cards are dealt from a 52-card deck without replacement. What is the theoretical probability all three are hearts?

Example 5

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Two fair coins are tossed. Find the theoretical probability of getting exactly one head.

Example 6

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A fair die and a fair coin are used together. What is the probability of rolling a 3 AND getting heads?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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A fair die is rolled. What is the theoretical probability of rolling an odd number?

Example 2

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

Example 3

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A standard die is rolled. What is the theoretical probability of rolling a 1 or a 2?

Example 4

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A spinner has 5 equal sections of different colors. What is the theoretical probability of landing on any one specific color?

Example 5

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A bag has 10 equally likely tokens, 7 of which are winning. What is the theoretical probability of drawing a winner?

Example 6

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What is the theoretical probability of rolling a 7 on a single standard die?

Example 7

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A card is drawn from a 52-card deck. What is the theoretical probability of drawing an ace?

Example 8

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Two fair coins are flipped. Theoretically, what is the probability of getting two heads?

Example 9

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A die is rolled. What is the theoretical probability of rolling a number that is both even and greater than 3?

Example 10

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A spinner has 12 equal sections numbered 1 to 12. What is the theoretical probability of landing on a factor of 12?

Example 11

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Two dice are rolled. What is the theoretical probability that the sum is 7?

Example 12

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A box has 5 red, 3 green, and 2 blue balls. What is the theoretical probability of drawing a ball that is not green?

Example 13

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Two dice are rolled. What is the theoretical probability that both show the same number?

Example 14

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A spinner has sections of probability 12\frac{1}{2} (win) and 12\frac{1}{2} (lose). Theoretically, in 50 spins how many wins are expected?

Example 15

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A card is drawn from a 52-card deck. What is the theoretical probability of drawing a face card (J, Q, K)?

Example 16

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Two dice are rolled. What is the theoretical probability that the product of the two numbers is even?

Example 17

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A bag has tiles numbered 1 to 9. What is the theoretical probability of drawing a perfect square?

Example 18

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Two dice are rolled. What is the theoretical probability that the sum is greater than 9?

Example 19

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A bag has tickets numbered 1 to 30. What is the theoretical probability of drawing a number divisible by 3 or by 5?

Example 20

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Three fair coins are flipped. What is the theoretical probability of getting exactly two heads?

Example 21

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A bag has 4 red, 3 blue, and 1 green marble. What is the theoretical probability of drawing a blue marble?

Example 22

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A spinner is divided into 8 equal sections numbered 1 to 8. What is the theoretical probability of landing on a prime number?

Example 23

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A jar holds 20 jelly beans: 8 yellow, 5 red, 4 green, 3 black. What is the theoretical probability of picking a red one?

Example 24

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A standard die is rolled. What is the theoretical probability of rolling a number less than 5?

Example 25

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A drawer has 6 black socks and 4 white socks. What is the theoretical probability of drawing a black sock at random?

Example 26

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Two fair coins are flipped. What is the theoretical probability of getting exactly one head?

Example 27

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Two standard dice are rolled. What is the theoretical probability that the sum is 10?

Example 28

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A card is drawn from a 52-card deck. What is the theoretical probability that it is a face card (J, Q, or K)?

Example 29

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A spinner has 10 equal sections numbered 1 to 10. What is the theoretical probability of landing on a multiple of 3?

Example 30

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A standard die is rolled. What is the theoretical probability of rolling a number that is a perfect square?

Example 31

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Two fair coins are flipped. What is the theoretical probability of getting at least one tail?

Example 32

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Two dice are rolled. What is the theoretical probability that the product of the two faces is even?

Example 33

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Three fair coins are flipped. What is the theoretical probability of getting exactly two heads?

Example 34

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Two dice are rolled. What is the theoretical probability that the sum is greater than 9?

Example 35

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A card is drawn from a 52-card deck. What is the theoretical probability that it is a red face card?

Example 36

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Two dice are rolled. What is the theoretical probability that the two faces show the same number?

Example 37

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Two dice are rolled. What is the theoretical probability that the larger face (or the common face if doubles) is exactly 4?

Example 38

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A bag has 4 red and 6 blue balls. One ball is drawn, replaced, and a second is drawn. What is the probability both are red?

Example 39

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Two fair six-sided dice are rolled. What is the theoretical probability that the sum is 7?
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Background Knowledge

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

probability basicrelative frequency