Exercise 23 Page 843

To find the similarities and differences between theoretical and experimental probabilities, let's first analyze an example experiment. Consider the experiment of flipping a fair coin and rolling a fair die.

Suppose we want to find the probability of getting a head on the coin and a $1$ on the die. We will find the theoretical and experimental probabilities of this event first.

Theoretical Probability

The theoretical probability describes the likelihood of an event based on mathematical reasoning. It is given by the ratio of the number of favorable outcomes to the number of possible outcomes.

P (event) = \frac{Number of favorable outcomes}{Number of possible outcomes}

The possible outcomes are given by combining the possible outcomes on the coin and the possible outcomes on the die. Let's list them.

Possible Outcomes
$(Head, 1)$	$(Tail, 1)$
$(Head, 2)$	$(Tail, 2)$
$(Head, 3)$	$(Tail, 3)$
$(Head, 4)$	$(Tail, 4)$
$(Head, 5)$	$(Tail, 5)$
$(Head, 6)$	$(Tail, 6)$

Note that there are

12

possible outcomes and only

one

way to get

(Head, 1) .

Therefore, the probability of getting a head on the coin and

1

on the die is given by the ratio of

1

to

12 .

P (Head and 1) = \frac{1}{1 2}

Experimental Probability

The experimental probability of an event

A

measures the likelihood of an event based on the actual results of an experiment. It is given by the ratio of the number of times the event occurs to the number of trials.

P (A) = \frac{Number of times the event occurs}{Number of trials}

Now, suppose that we conducted the experiment and recorded the results of

15

trials in a table.

Flipping a Coin and Rolling a Die $15$ Times
Number of Trial	Coin	Die	Head and $1$ ?
$1$	Head	$3$	$\times$
$2$	Head	$5$	$\times$
$3$	Tail	$2$	$\times$
$4$	Head	$1$	$✓$
$5$	Head	$4$	$\times$
$6$	Tail	$1$	$\times$
$7$	Head	$6$	$\times$
$8$	Tail	$1$	$\times$
$9$	Head	$3$	$\times$
$10$	Tail	$4$	$\times$
$11$	Head	$4$	$\times$
$12$	Tail	$5$	$\times$
$13$	Tail	$1$	$\times$
$14$	Tail	$6$	$\times$
$15$	Head	$3$	$\times$

We can see that there was only

one

trial out of

15

where we got a head on the coin and a

1

on the die. Therefore, by calculating this ratio, we will find the experimental probability.

P (Head and 1) = \frac{1}{1 5}

Comparison

Comparing the probabilities, we can see that the experimental probability is different from what we expected to happen. However, if we conduct a more significant number of trials, these probabilities will eventually become close to each other. With this information, we can make some conclusions about the similarities and differences.

How are the Probabilities Similar? The ratios of both probabilities are defined similarly. For theoretical probability, the ratio is the number of favorable outcomes to the number of possible outcomes. Conversely, the experimental probability is the ratio of the number of successes to the number of trials.
How are the Probabilities Different? Theoretical probability is what we expect to happen based on mathematical reasoning. Conversely, the experimental probability is what happens based on the actual results of an experiment.

Please note that experiments may vary. Depending on the number of trials, probabilities will be closer to each other.