THEORETICAL vs EXPERIMENTAL PROBABILITY

Theoretical Probability is the theory behind probability.

Experimental (empirical) probability is the probability calculated during experiments, direct observation, experience, or practice.

With theoretical probability, we don’t actually conduct an experiment (i.e. roll a die or conduct a survey). Instead, we use our knowledge about a situation, some logical reasoning, and/or known formula to calculate the probability of an event happening.

Example :

Let us compare theoretical and experimental probabilities with the experiment of rolling a six faced cube (faces numbered from 1 to 6) for 30 times.

Step 1 :

When we roll the cube once, the theoretical probabilities for different outcomes. 

There are 18 possible outcomes in the sample space.

Getting number 1 ---> Theoretical probability  =  1/6

Getting number 2 ---> Theoretical probability  =  1/6

Getting number 3 ---> Theoretical probability  =  1/6

Getting number 4 ---> Theoretical probability  =  1/6

Getting number 5 ---> Theoretical probability  =  1/6

Getting number 6 ---> Theoretical probability  =  1/6

Step 2 : 

When we roll the cube for 30 times, number of times each number will be rolled.

1 : 30 x 1/6  =  5 times

2 : 30 x 1/6  =  5 times

3 : 30 x 1/6  =  5 times

4 : 30 x 1/6  =  5 times

5 : 30 x 1/6  =  5 times

6 : 30 x 1/6  =  5 times

Step 3 : 

When we roll the cube for 30 times, number of times each number will be rolled.

Number 1 : 30 x 1/6  =  5 times

Number 2 : 30 x 1/6  =  5 times

Number 3 : 30 x 1/6  =  5 times

Number 4 : 30 x 1/6  =  5 times

Number 5 : 30 x 1/6  =  5 times

Number 6 : 30 x 1/6  =  5 times

According to theoretical probability, we get each number 5 times in rolling the cube 30 times.

Step 4 : 

Now, let us do the actual experiment and record the results. That is, rolling a cube 30 times. 

From the above table, after simplification, we get the following experimental probabilities for each number. 

Getting number 1 ---> Experimental probability  =  1/5

Getting number 2 ---> Experimental probability  =  2/15

Getting number 3 ---> Experimental probability  =  1/10

Getting number 4 ---> Experimental probability  =  1/5

Getting number 5 ---> Experimental probability  =  7/30

Getting number 6 ---> Experimental probability  =  2/15

Step 5 : 

Look at the results in step 1 and step 4.

How do the experimental probabilities compare with the theoretical probabilities ?

The experimental probabilities are not close to the theoretical probabilities.

But, by performing more trials, we may tend to get experimental results that are closer to the theoretical probabilities.

Reflect

Could the experimental probabilities ever be exactly equal to the theoretical probability? If so, how likely is it? If not, why not ?

Yes, the experimental probabilities could be exactly equal to the theoretical probabilities, but it is very unlikely.

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