075. Experimental Probability and Expected Frequency

Learning Intentions

• To understand what experimental probability is and how it is related to the theoretical probability of an event for a large number of trials
• find the expected number of occurrences of an event
• find the experimental probability of an event given experiment results

Pre-requisite Summary

• Understand that probability is a number between $0$ and $1$ inclusive
• Know that theoretical probability is found using equally likely outcomes
• Understand that an experiment involves repeated trials
• Know that an outcome is the result of one trial and an event is a set of outcomes
• Be able to write probability as a fraction, decimal or percentage
• Be able to multiply a probability by a number of trials
• Understand that larger numbers of trials usually make experimental probability closer to theoretical probability

Worked Examples

Worked Example 1

A fair coin is tossed many times.
a) Explain what experimental probability means.
b) State the theoretical probability of heads.
c) Explain how the experimental probability of heads is expected to behave as the number of tosses becomes very large.

Worked Example 2

A fair die is rolled $60$ times.
a) Find the theoretical probability of rolling a $4$.
b) Find the expected number of times a $4$ should occur.
c) Explain why the actual number rolled in an experiment may be different.

Worked Example 3

A spinner lands on red $18$ times in $50$ spins.
a) Find the experimental probability of landing on red.
b) Write the answer as a fraction, decimal and percentage.
c) Explain whether the result suggests red happened less than, equal to, or more than half the time.

Worked Example 4

A bag contains $3$ blue counters and $1$ yellow counter. One counter is chosen, replaced, and this is repeated $40$ times.
a) Find the theoretical probability of yellow.
b) Find the expected number of yellow results in $40$ trials.
c) If yellow occurs $7$ times, find the experimental probability of yellow.

Worked Example 5

A fair coin is tossed $200$ times and lands heads $112$ times.
a) Find the experimental probability of heads.
b) Compare it with the theoretical probability.
c) Explain why the two probabilities are close but not exactly equal.

Worked Example 6

A game has theoretical probability ${\displaystyle \frac{3}{10}}$ of winning. It is played $150$ times.
a) Find the expected number of wins.
b) If the game is actually won $51$ times, find the experimental probability of winning.
c) Compare the experimental probability with the theoretical probability.

Problems

Problem 1

A fair coin is tossed many times.
a) Explain what experimental probability means.
b) State the theoretical probability of tails.
c) Explain how the experimental probability of tails is expected to behave as the number of tosses becomes very large.

Problem 2

A fair die is rolled $48$ times.
a) Find the theoretical probability of rolling a $2$.
b) Find the expected number of times a $2$ should occur.
c) Explain why the actual number rolled in an experiment may be different.

Problem 3

A spinner lands on blue $21$ times in $60$ spins.
a) Find the experimental probability of landing on blue.
b) Write the answer as a fraction, decimal and percentage.
c) Explain whether the result suggests blue happened less than, equal to, or more than half the time.

Problem 4

A bag contains $4$ green counters and $2$ red counters. One counter is chosen, replaced, and this is repeated $30$ times.
a) Find the theoretical probability of red.
b) Find the expected number of red results in $30$ trials.
c) If red occurs $8$ times, find the experimental probability of red.

Problem 5

A fair coin is tossed $120$ times and lands heads $67$ times.
a) Find the experimental probability of heads.
b) Compare it with the theoretical probability.
c) Explain why the two probabilities are close but not exactly equal.

Problem 6

A game has theoretical probability ${\displaystyle \frac{2}{5}}$ of winning. It is played $100$ times.
a) Find the expected number of wins.
b) If the game is actually won $37$ times, find the experimental probability of winning.
c) Compare the experimental probability with the theoretical probability.

Exercises

Understanding and Fluency

1. State whether each probability is theoretical or experimental:
   a) ${\displaystyle \frac{1}{2}}$ for heads on a fair coin
   b) ${\displaystyle \frac{23}{50}}$ for heads after $50$ tosses
   c) ${\displaystyle \frac{1}{6}}$ for rolling a $6$ on a fair die

2. Explain the meaning of each term:
   a) theoretical probability
   b) experimental probability
   c) expected number of occurrences

3. Find the expected number of occurrences:
   a) probability ${\displaystyle \frac{1}{2}}$ in $20$ trials
   b) probability ${\displaystyle \frac{1}{4}}$ in $40$ trials
   c) probability ${\displaystyle \frac{3}{5}}$ in $50$ trials

4. Find the expected number of occurrences:
   a) probability ${\displaystyle \frac{1}{6}}$ in $60$ trials
   b) probability ${\displaystyle \frac{2}{3}}$ in $30$ trials
   c) probability ${\displaystyle \frac{3}{10}}$ in $200$ trials

5. Find the experimental probability from the results:
   a) $12$ successes in $20$ trials
   b) $7$ successes in $25$ trials
   c) $31$ successes in $50$ trials

6. Find the experimental probability from the results:
   a) $18$ successes in $40$ trials
   b) $45$ successes in $100$ trials
   c) $9$ successes in $12$ trials

7. Write each experimental probability as a fraction, decimal and percentage:
   a) $14$ successes in $20$ trials
   b) $16$ successes in $50$ trials
   c) $72$ successes in $120$ trials

8. Compare theoretical and experimental probability:
   a) theoretical ${\displaystyle \frac{1}{2}}$, experimental ${\displaystyle \frac{27}{50}}$
   b) theoretical ${\displaystyle \frac{1}{6}}$, experimental ${\displaystyle \frac{8}{36}}$
   c) theoretical ${\displaystyle \frac{3}{4}}$, experimental ${\displaystyle \frac{70}{100}}$

Reasoning

9. Explain why experimental probability can be different from theoretical probability.

10. A student says that if the theoretical probability of an event is ${\displaystyle \frac{1}{4}}$, then the event must happen exactly $25$ times in $100$ trials. Explain why this is incorrect.

11. Explain why experimental probability usually gets closer to theoretical probability when the number of trials increases.

12. A student finds the experimental probability by dividing the number of failures by the total number of trials, even though the event is success. Explain the mistake.

Problem-solving

13. A fair die is rolled $90$ times.
    a) Find the expected number of times a $5$ should occur.
    b) If a $5$ actually occurs $12$ times, find the experimental probability.
    c) Compare it with the theoretical probability.

14. A spinner has theoretical probability ${\displaystyle \frac{3}{8}}$ of landing on green. It is spun $80$ times.
    a) Find the expected number of green results.
    b) If green occurs $34$ times, find the experimental probability.
    c) State whether the experimental probability is greater or less than the theoretical probability.

15. A bag contains $5$ black counters and $5$ white counters. One counter is chosen with replacement $60$ times. White occurs $28$ times.
    a) Find the theoretical probability of white.
    b) Find the expected number of white results.
    c) Find the experimental probability of white.

16. A game has probability ${\displaystyle \frac{1}{5}}$ of winning. It is played $150$ times and won $26$ times.
    a) Find the expected number of wins.
    b) Find the experimental probability of winning.
    c) Compare the two probabilities.

17. A coin is tossed $300$ times and lands tails $149$ times.
    a) Find the experimental probability of tails.
    b) Compare it with the theoretical probability.
    c) Explain why the result is reasonable.

18. A survey says a machine produces faulty parts with theoretical probability ${\displaystyle \frac{1}{20}}$. In $200$ parts, $14$ are faulty.
    a) Find the expected number of faulty parts.
    b) Find the experimental probability of a faulty part.
    c) Decide whether the machine produced more or fewer faulty parts than expected.

Potential Misunderstandings

• Students may confuse theoretical probability with experimental probability
• Students may think experimental probability must exactly equal theoretical probability
• Students may forget that expected number means probability multiplied by the number of trials
• Students may calculate experimental probability using the wrong number of outcomes
• Students may think a large number of trials guarantees exact agreement with theoretical probability
• Students may compare probabilities in different forms without converting them carefully
• Students may confuse the number of occurrences with the probability itself
• Students may not recognise that replacement is important for keeping the theoretical probability constant across trials