Unit 6 Section 4 : Theoretical and Experimental Probabilities

In this section we compare theoretical and experimental probabilities.

The term 'theoretical probabilities' describes those which have been calculated, for example by the methods described in sections 6.2. and 6.3.

'Experimental probabilities' are estimates for probabilities that cannot be determined logically. They can be derived from the results of experiments, but often they are obtained from the analysis of statistical data or historical records.

Here we obtain experimental probabilities from simple experiments and compare them with the theoretical probabilities.

Example 1

An unbiased dice is to be rolled 240 times.

(a)

Calculate the number of times you would expect to obtain each of the possible scores.

p(6) =
Expected number of 6s = × 240 = 40
Similarly, you would expect to obtain each of the possible scores 40 times.
(b)

Now roll the dice 240 times and collect some experimental results, presenting them in a bar chart.

The results of the experiment are recorded in the following table:
These results are illustrated in the following bar chart. A horizontal line has been drawn to show the expected frequencies for the scores.

Note that none of the bars is of the expected height; some are above and some are below. However, all the bars are close to the predicted number.
We would not expect to obtain exactly the predicted number. The more times the experiment is carried out, the closer the experimental results will be to the theoretical predictions.

Exercises

Question 1

(a)

A fair coin is tossed 100 times. How many heads and how many tails would your expect to obtain?

heads and tails
(b)

Toss a fair coin 100 times and display your results using a bar chart.

Results:
Heads =
Tails =
(c)

Compare your theoretical predictions with your experimental results.

Heads are ( by )
and tails are ( by )
than the theoretical predictions.
Question 2

Two fair coins are to be tossed at the same time.

(a)

Calculate the probability of obtaining:

(i) 2 heads, (ii) a head and a tail, (iii) 2 tails.
(b)

Calculate the number of times you would expect to obtain each outcome if the coins are tossed 100 times.

times times times
(c)

Toss two coins 100 times and illustrate your results using a bar chart.

Results:
Heads =
Heads and tails =
Tails =
(d)

Compare your theoretical predictions with your experimental results.

HH: ( by )
HT: ( by )
TT: ( by )
Question 3

(a)

List the 8 possible outcomes when 3 fair coins are tossed at the same time.

(b)

If three fair coins were tossed 32 times, how many times would you expect to obtain:

(i) 3 heads,
(ii) 2 heads,
(iii) 1 head,
(iv) 0 heads ?
(c)

Carry out an experiment and compare your theoretical predictions with your experimental results.

Results:
HHH =
HHT =
HTT =
TTT =
Compare:
HHH: ( by )
HHT: ( by )
HTT: ( by )
TTT: ( by )
Question 4

(a)

What are the expected frequencies of the totals 2, 3, 4, ..., 11, 12 when two fair dice are thrown at the same time and the experiment is repeated 36 times?

Total
Expected Frequency
(b)

Carry out the experiment in (a) and compare the predicted and experimental frequencies.

Total
Experiment
Difference
(c)

Repeat (a) and (b) for 144 throws.

Total
Expected Frequency
Experiment
Difference
Question 5

A fair coin and an unbiased dice are thrown at the same time. A score is then calculated using the following rules:

(a)

Use a two-way table to show all the possible scores.

Dice
Coin
1 2 3 4 5 6
Heads
Tails
(b)

Draw up a table showing the theoretical probabilities for the various scores.

Total
Theoretical Probability
(c)

If the coin and the dice are thrown 120 times, how many times would you expect to obtain each score?

Total
Expected Frequency
(d)

Conduct an experiment and compare your experimental results with your answers to part (c).

Experiment
Difference
Question 6

A dice with 4 faces has one blue, one green, one red and one yellow face.
Five pupils did an experiment to investigate whether the dice was biased or not.

The following table shows the data they collected.

(a)

Which pupil's data is most likely to give the best estimate of the probability of getting each colour on the dice?

Because greater number of trials leads to better results.
The pupils collected all the data together.
(b)

Consider the data. Choose whether you think the dice is biased or unbiased, and explain your answer.

The data does indicate that the dice is biased. If it was unbiased, we would expect to get frequencies of approximately 130 for each colour. However, the frequencies for red and blue are well above 130 and the frequencies for green and yellow are well below 130, which is fairly strong evidence that the dice is biased.
(c)

From the data, work out the probability of the dice landing on the blue face.

p(B) = =
(d)

From the data work out the probability of the dice landing on the green face.

p(G) = =
Question 7

Some pupils threw 3 fair dice. They recorded how many times the numbers on the dice were the same.

(a)

Choose the name of the pupil whose data are most likely to give the best estimate of the probability of getting each result.

Sue, because she conducted the experiment the greatest number of times.
(b)

This table shows the pupils' results collected together:

Use these data to estimate the probability of throwing numbers that are all different.

= =
(c)

The theoretical probability of each result is shown below:

Use these probabilities to calculate, for 300 throws, how many times you would theoretically expect to get each result. Complete the table below.