In this lesson we
will examine two different types of probability,
empirical
and
theoretical
probability.
Although both of these methods for determining the probability of an
event involve the concept of "chance", the solutions are
obtained in two very different ways.


Empirical Probability
When you perform
an experiment, or sample a number of people or objects to
determine the probability of an event, you are determining the
"empirical probability" of an event.
For example, let's say that a manufacturer tested 1000 radios,
at random, and found 15 of them to be defective.
We can easily determine that the empirical probability that a
radio is defective would be:
P(defective radio) = 15
1000
or...3/200.
As a decimal it would be .015, and
as a percent it would be 1.5%
Now the manufacturer can use this result to
predict that in the
production of 7500 radios, 1.5% of them will
probably be defective. Or,
(.015)(7500) = 112.5 defective radios.
Let's look at another example of
empirical probability.
Let's say that Sheila tossed a coin 200 times and recorded
whether it was a "head", or a "tail" after each toss.
After the experiment, Sheila found that she had recorded 122
"heads" and 78 "tails."
From this experiment she could predict that the
empirical probability
of getting a "head" is:
P(Head) = 122
200
or..61/100.
As a decimal it would be .61, and as a percent 61%.
Because in determining empirical probability you actually
perform an experiment, it is sometimes called:
"experimental probability."

Theoretical
Probability
An empirical approach is required whenever we want to determine
the probability of an event by actually performing the
experiment. However, common sense tells us that there is
also a simple, mathematical, way to determine the probability of
an event in another way.
For example, Jack is playing a game in which each player must
roll a die. To win the game Jack must roll a number
greater than 4. What is the probability that Jack will win
on his next turn?
Common sense tells us that:
(a) The die has an equal chance of landing in 6 ways.
(b) There are only 2 ways for Jack to win the game.
(c) Therefore the probability of Jack winning is:
P(Jack wins) = 2
6
or...1/3.
As a decimal it would be .333..., as a percent, 33 1/3%
And one
more example of "theoretical probability."
In this game a standard 52 card deck of playing cards is used.
In order to win you must pick a "face card."
What is the probability that you will win the
game on the next draw?
Again, common sense tells us:
(a) Each card in the deck has an equal chance of being drawn.
(b) There are 12 face cards (winning cards) in the deck.
Therefore the probability of winning on the next draw is:
P(face card) = 12
52
or... 3/13.
As a decimal it would be about .23, and as a percent, 23%

An
empirical
example
During the last 15 basketball games, Sam has made 64 and
missed 32 foul shots. What is the empirical (or
experimental) probability that Sam will make his next foul
shot?

Using
the statistical information in the problem we can determine that
the empirical probability of Sam making his next foul shot would
be:
P(making the next shot) = 64 (made)
96 (total)
or..2/3.
as a decimal it would be .666..., as a percent, 66 2/3%

A
theoretical
example
In a box there are 4 red marbles, 3 blue marbles and 5
yellow marbles. If one marble is chosen at random from
the box, what is the probability of choosing:
(a) a red marble
(b) a blue or yellow marble

Remember...don't forget your
"common sense!"
P(red) = 4
12
or 1/3, (decimal):.333.., (percent): 33 1/3%
P(blue or yellow) = 8
12
or 2/3, (decimal):.666.., (percent): 66 2/3%

