Probability: theoretical & experimental

Theoretical probability predicts the chance of an outcome using maths, before anything happens — like knowing a coin has a 1-in-2 (or 1/2) chance of landing on heads because there are 2 equally likely outcomes. The sample space is the full list of all possible outcomes, like {Heads, Tails} for a coin, or {1,2,3,4,5,6} for a die. Experimental probability comes from actually running trials and recording results — like flipping a coin 100 times and counting how many were heads. With enough trials, experimental results tend to get closer to the theoretical prediction, but they won't always match exactly, especially with small numbers of trials.

Key terms

Sample space:

The full list of all possible outcomes of an event.

Theoretical probability:

The predicted chance of an outcome, calculated using maths.

Experimental probability:

The chance of an outcome based on actually running trials and recording results.

Questions

1. 1. The sample space is:

   • One possible outcome
   • The full list of all possible outcomes
   • The probability itself
   • A type of graph

2. 2. The sample space for a coin flip is:

   • {Heads}
   • {Heads, Tails}
   • {1,2,3}
   • {Tails, Tails}

3. 3. The sample space for a standard die roll is:

   • {1,2,3}
   • {1,2,3,4,5,6}
   • {1,6}
   • {Heads, Tails}

4. 4. Theoretical probability is calculated:

   • By running experiments only
   • Using maths, before anything happens
   • By guessing
   • Randomly

5. 5. Experimental probability comes from:

   • Maths only
   • Actually running trials and recording results
   • Guessing
   • The sample space alone

6. 6. The theoretical probability of flipping heads on a fair coin is:

   • 1/4
   • 1/3
   • 1/2
   • 1

7. 7. The theoretical probability of rolling a 6 on a fair die is:

   • 1/2
   • 1/3
   • 1/6
   • 1

8. 8. If you flip a coin 100 times, roughly how many heads would you theoretically expect?

   • 25
   • 50
   • 75
   • 100

9. 9. If you roll a die 60 times, roughly how many times would you expect to roll a 3?

   • 6
   • 10
   • 20
   • 30

10. 10. Experimental results with only a few trials:

    • Always exactly match theoretical probability
    • May differ from theoretical probability just by chance
    • Are always more accurate than theory
    • Cannot be recorded

11. 11. As the number of trials increases, experimental probability tends to:

    • Move further from the theoretical value
    • Get closer to the theoretical value
    • Stay exactly the same always
    • Become impossible to calculate

12. 12. The probability of an impossible event is:

    • 0
    • 0.5
    • 1
    • Undefined

13. 13. The probability of a certain event is:

    • 0
    • 0.5
    • 1
    • Undefined

14. 14. A bag has 3 red balls and 2 blue balls. What is the theoretical probability of picking a red ball?

    • 2/5
    • 3/5
    • 1/2
    • 3/2

15. 15. A coin is flipped 10 times and lands heads 7 times. Does this mean the coin is unfair?

    • Yes, definitely
    • Not necessarily — small samples can vary by chance
    • Yes, since it should always be exactly 5
    • No, coins can never be unfair

16. 16. A die is rolled 600 times. Theoretically, about how many times would you expect a 4?

    • 50
    • 100
    • 150
    • 200

17. 17. A spinner has 4 equal sections: red, blue, green, yellow. What is the probability of landing on blue OR green?

    • 1/4
    • 1/2
    • 3/4
    • 1

18. 18. In 50 trials of flipping a coin, you get 32 heads. What is the experimental probability of heads from this data?

    • 0.32
    • 0.5
    • 0.64
    • 0.18

19. 19. Why might a small number of trials give a misleading picture of an event's true probability?

    • Small samples always show the exact true probability
    • Random chance can cause noticeable variation with few trials
    • Small samples remove all randomness
    • Probability doesn't apply to small samples

20. 20. A bag has 10 balls: some red, some blue. After 100 draws (with replacement), 68 were red. What is the best estimate of the probability of drawing red?

    • 0.10
    • 0.32
    • 0.68
    • 1.00