Open Math›Year 9›Probability›Experimental probability

Experimental probability

Calculate relative frequencies from experiments, compare experimental and theoretical probability, and predict expected frequencies from experimental data.

Worked examples

Calculating relative frequency

Straightforward

Problem

A spinner is spun

50

times. It lands on blue

15

times. Find the experimental probability of landing on blue.

Identify the number of trials and the frequency of the event

Total number of spins (trials)

= 50

. Number of times blue appeared (frequency)

= 15

Apply the experimental probability formula

P(\text{blue}) = \frac{\text{frequency of event}}{\text{total number of trials}} = \frac{15}{50}

Simplify the fraction

\frac{15}{50} = \frac{3}{10}

Answer

P(\text{blue}) = \dfrac{3}{10}

Comparing experimental and theoretical probability

Moderate

Problem

A fair six-sided die is rolled

60

times and a

1

appears

7

times. Find the experimental probability of rolling a

1

, and compare it to the theoretical probability.

Calculate the experimental probability

P_\text{exp}(1) = \frac{7}{60}

State the theoretical probability

For a fair die, each face is equally likely.

P_\text{theo}(1) = \dfrac{1}{6} = \dfrac{10}{60}

Compare the two values

Experimental:

\dfrac{7}{60} \approx 0.117

. Theoretical:

\dfrac{10}{60} \approx 0.167

. The experimental probability is less than the theoretical probability. This is normal — experimental results vary from trial to trial, and tend to get closer to the theoretical value as the number of trials increases.

Answer

P_\text{exp}(1) = \dfrac{7}{60}

, which is less than the theoretical probability of

\dfrac{1}{6}

Predicting expected frequency

Challenging

Problem

In an experiment, a bag of counters is sampled with replacement

60

times and a green counter is drawn

18

times. If the bag is sampled

300

times, how many times would you expect to draw green?

Find the experimental probability of drawing green

P(\text{green}) = \frac{18}{60} = \frac{3}{10}

Use the experimental probability to find the expected frequency

\text{Expected frequency} = P(\text{green}) \times \text{number of trials} = \frac{3}{10} \times 300

Calculate the result

\frac{3}{10} \times 300 = 90

Answer

You would expect to draw green

90

times.

Practise

Q1·Straightforward

A coin is flipped

20

times and lands on heads

8

times. What is the experimental probability of heads?

Explanation

Experimental probability

= \dfrac{8}{20} = \dfrac{2}{5}

Q2·Straightforward

A spinner is spun

50

times and lands on red

15

times. What is the experimental probability of landing on red?

Explanation

Experimental probability

= \dfrac{15}{50} = \dfrac{3}{10}

Q3·Straightforward

A die is rolled

30

times and shows an even number

12

times. What is the experimental probability of rolling an even number?

Explanation

Experimental probability

= \dfrac{12}{30} = \dfrac{2}{5}

Q4·Straightforward

A counter is drawn from a bag and replaced

40

times. Blue is drawn

10

times. What is the experimental probability of drawing blue?

Explanation

Experimental probability

= \dfrac{10}{40} = \dfrac{1}{4}

Q5·Moderate

A fair coin is tossed

100

times and lands on tails

47

times. What is the experimental probability of tails? Give your answer as a decimal.

Explanation

Experimental probability

= \dfrac{47}{100} = 0.47

. The theoretical probability is

0.5

, so the experimental result is slightly below the expected value.

Q6·Moderate

A fair six-sided die is rolled

60

times and a

5

appears

8

times. The theoretical probability of rolling a

5

\dfrac{1}{6}

. Is the experimental probability greater than, less than, or equal to the theoretical probability?

Greater than the theoretical probabilityLess than the theoretical probabilityEqual to the theoretical probabilityCannot be determined

Explanation

Experimental probability

= \dfrac{8}{60} = \dfrac{2}{15} \approx 0.133

. Since

0.133 < 0.167

, the experimental probability is less than the theoretical probability.

Q7·Moderate

A spinner with

4

equal sections (red, blue, green, yellow) is spun

200

times. The results are: red

52

, blue

49

, green

54

, yellow

45

. Which colour has an experimental probability closest to the theoretical probability of

\dfrac{1}{4}

RedBlueGreenYellow

Explanation

Theoretical probability

= 0.25

. Experimental probabilities: Red

= \frac{52}{200} = 0.26

(difference

0.01

), Blue

= \frac{49}{200} = 0.245

(difference

0.005

), Green

= \frac{54}{200} = 0.27

(difference

0.02

), Yellow

= \frac{45}{200} = 0.225

(difference

0.025

). Blue is closest.

Q8·Moderate

A fair coin is tossed in two trials. Trial 1:

10

tosses, heads appears

4

times. Trial 2:

1000

tosses, heads appears

487

times. The theoretical probability of heads is

0.5

. Which trial gives an experimental probability closer to

0.5

Trial 1 (10 tosses)Trial 2 (1000 tosses)Both are equally closeCannot be determined

Explanation

Trial 1:

\frac{4}{10} = 0.4

, difference from

0.5

0.1

. Trial 2:

\frac{487}{1000} = 0.487

, difference from

0.5

0.013

. Trial 2 is much closer. In general, experimental probability tends to get closer to the theoretical probability as the number of trials increases.

Q9·Challenging

A biased coin shows heads with experimental probability

\dfrac{3}{5}

. If the coin is tossed

200

times, how many times would you expect heads to appear?

Explanation

Expected frequency

= \dfrac{3}{5} \times 200 = 120

times.

Q10·Challenging

A spinner is spun

40

times during an experiment and lands on yellow

10

times. If the spinner is spun

240

times, how many times would you expect it to land on yellow?

Explanation

Experimental probability of yellow

= \dfrac{10}{40} = \dfrac{1}{4}

. Expected frequency

= \dfrac{1}{4} \times 240 = 60

times.

Q11·Challenging

A six-sided die is rolled

50

times and a

2

appears

8

times. Based on this experimental probability, how many times would you expect a

2

to appear if the die is rolled

300

times?

Explanation

Experimental probability of rolling a

2 = \dfrac{8}{50} = \dfrac{4}{25}

. Expected frequency

= \dfrac{4}{25} \times 300 = 48

times.

Q12·Challenging

A bag contains red and blue counters. In

25

draws with replacement, red is drawn

10

times. Based on this experiment, how many times would you expect to draw red in

150

draws?

Explanation

Experimental probability of red

= \dfrac{10}{25} = \dfrac{2}{5}

. Expected frequency

= \dfrac{2}{5} \times 150 = 60

times.