Experimental probability is defined as the relative frequency of an event of interest,
or the number of times that particular event occurred divided by the total number of times
the experiment was repeated.
Experimental probability needs an experiment to be done first, and then you can calculate
the relative frequency of the outcome you are interested in.
But if you repeat your experiment a large enough number of times, something very interesting
happens to the experimental probability.
Suppose I roll a normal six-sided die.
There are six possible outcomes, numbered one through to six.
I might do the experiment ten times, ten rolls, and get a distribution like this.
The graph shows the relative frequency of each outcome, as my experiment progresses.
But suppose I repeat the experiment a whole lot more times.
After fifty rolls, the results look like this.
After two hundred rolls, like this.
And after a thousand rolls...
Notice how the relative frequency of all the possible outcomes has become basically the
same.
If I repeat the experiment often enough, the experimental probability of any one outcome
becomes closer and closer to a particular fixed number.
We call that number the theoretical probability.
The theoretical probability of an event is simply a prediction, in advance, of the expected
relative frequency of that particular event, if the experiment is repeated a large number
of times.
And the most remarkable thing is that, for many simple experiments, you can predict very
well (and very simply) the precise value of that probability.
If you know the sample space, the set of all possible outcomes, and the outcomes are all
equally likely, then the theoretical probability is just the number of favourable outcomes
divided by the total size of the sample space.
The predicted relative frequency is the number of outcomes that match your particular event,
divided by the total number of possible outcomes in the sample space.
So for that die rolling experiment, the theoretical probability of rolling, say, a three, should
be the number of threes on the die, which is one, out of the number of possible outcomes,
which is six.
One in six.
That's zero point one six recurring.
Let's have another look at that graph.
Yes, that's about the value that all the experimental probabilities are getting closer and closer
to.
Good.
What if instead I'm interested in more than one outcome?
What's the theoretical probability of rolling a number less than three?
Well, there are two outcomes less than three, out of the six possible outcomes.
So the theoretical probability is one in three, or thirty-three and a third percent.
Now you could roll a real die many times, and not get a number less than three.
You could get an experimental probability less or greater than one-third.
But if you roll the die enough times, your experimental probability, the relative frequency
of rolling a number less than three, will gradually get closer and closer to this value.
Experimental probability is based on the actual outcomes after doing an experiment.
Theoretical probability is based on the sample space alone.
It's a prediction of what you expect to happen if you do the experiment a sufficiently large
number of times.
Sometimes you have to be careful about the way you count the sample space.
Suppose I have a bag with some coloured marbles in it.
Let's say there are three red marbles and five blue marbles.
I'm thinking of performing an experiment where I take one marble out of the bag at a time.
I'll put the marble I get out back in again before taking the next.
What's the theoretical probability of drawing out a red marble?
Well, what are the possible outcomes?
What does the sample space look like?
It's tempting to say the outcomes are "red" or "blue": two possible outcomes.
But actually that's not enough.
Remember there are three possible red marbles, and five possible blue ones.
That's a total of eight possible outcomes in my sample space.
Of those eight, three of them are red.
So the theoretical probability of drawing a red marble is three out of eight, or thirty-seven
and a half percent.
And of course you can use theoretical probability to predict the expected frequency of an event,
in exactly the same way as with experimental probability.
The expected frequency is the probability multiplied by the number of trials.
Take our bag of marbles again.
If I were to draw out a marble at random forty-six times, how many red marbles will I get?
Remember, we decided that the theoretical probability of getting a red marble was three
out of eight.
So the number of red marbles I should expect to get over forty-six trials is just three-eighths
of forty-six, which is around seventeen times, to the nearest whole number.
I would expect to get a red marble out of the bag about seventeen times out of forty-six.