Experimental Probability verses Theoretical Probability
Experimental Probability aka empirical probability: based on the outcomes of an actual experiment.
Game: Rock, Paper, Scissors
Rock crushes paper, scissors cut paper, and paper covers rock. If each
person playing creates the same gesture there is a tie and the game is
played again.
Johnny and Lulu like to play the game of Rock, Paper, Scissors, to
decide where they are going to eat dinner. Lulu’s last “showdown” with
Johnny is as follows, five paper, two rocks, and three scissors. The
results are seen in the following graph.
Frequency
Rock, Paper, Scissors
Outcome/Frequency
6
5
4
3
2
1
0
Paper
Rock
Scissors
Outcomes
To determine the probability of a particular event, you divide the number of times a particular outcome
5
1
or ; the
occurs by the number of trials. In the example, the probability that Lulu will throw paper is
10 2
3
2
1
or . The
probability that she will throw scissors is
; and the probability that she will throw rock is
10
10 5
probability examples might also be written as follows:
1
, where E represents the event of throwing “paper”
2
1
p(R) = , where R represents the event of throwing “rock”
5
3
p(S) = , where S represents the event of throwing “scissors”
10
p(E) =
The letter “E” was chosen to represent “paper” rather than the letter “P” because the letter “P” usually
represents probability in statistics, to avoid confusion, it is not used as a variable.
We are going to look at a larger sample set because ten is a small number of trials to base prediction. The
following graph looks at Lulu’s last 30 games of Rock, Paper, Scissors.
Frequency of Outcomes in
Lulu's last 30 games of Rock,
Paper, Scissors
Frequency
15
10
5
0
Paper
Rock
Scissors
Outcomes
After 30 throws, the probability of each event is as follows:
p(E) =
11
30
p(R) =
9
30
p(S) =
10
30
Notice that, as the number of trials increases, the differences between the probability values for each
1
possible outcome decreases. All are about pretty close to . This larger sample of throws better reflects
3
1
the true probabilities of each outcome, which in Lulu’s case appear to be about equal at each.
3
Theoretical Probability
Assuming that a rock, paper, scissors player like Lulu does not have a preference for any rock, paper, or
scissors. There is only one what to throw a rock, but there are three possible outcomes altogether.
1
Therefore, at any given time, you would predict that the probability that Lulu will throw a rock is .
3
Theoretical Probability is based on the mathematical analysis of probability that looks at all the possible
outcomes in an experiment.
For any situation involving probability, you can label the event of a successful outcome as A and the total
number of outcomes in the sample space as S. To determine the probability of a successful outcome, p(A),
n( A)
use the formula p( A) 
, where n(A) represents the number of ways a successful event can occur
n( S )
and n(S) represents the number of total possible outcomes in a probability experiment.
In any probability experiment, the probability values must lie between 0 and 1. If a particular outcome never
occurs, the probability that it will happen at any given time is zero.
If a particular outcome always occurs every time, it has a probability of 1.
In situations where particular outcomes occur rarely, occasionally, or often, they will have a theoretical
probability of somewhere between 0 and 1.
Example 1
(a) Name an event that would have a theoretical probability of 0.
__________________________________________________________________________________
__________________________________________________________________________________
(b) Name an event that would have a theoretical probability of 0.5
__________________________________________________________________________________
__________________________________________________________________________________
(c) Name an event that would have a theoretical probability of 1
__________________________________________________________________________________
__________________________________________________________________________________
Example 2
(a) Given that n(A) = 4 and n(S) = 20, determine p(A)
(b) Given that n(A) = 7 and p(A) = 0.1, determine n(S).
(c) Given that p(A) = 0.75 and n(S) = 60, determine n(A).