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)=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)=1/12
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)=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
*
2
Head
5
*
3
Tail
2
*
4
Head
1
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5
Head
4
*
6
Tail
1
*
7
Head
6
*
8
Tail
1
*
9
Head
3
*
10
Tail
4
*
11
Head
4
*
12
Tail
5
*
13
Tail
1
*
14
Tail
6
*
15
Head
3
*
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 and1)=1/15
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.
