When you gather data from observations during an experiment, you will be calculating an empirical (or experimental) probability.
The empirical (or experimental) probability of an event is an "estimate" that an event will occur based upon how often the event occurred after collecting data from an experiment in a large number of trials. This type of probability is based upon direct observations. Each observation in an experiment is called a trial.
Example: A survey was conducted to determine students' favorite brands of sneakers.
Each student chose only one brand from the list of brands A, B, C, D, or E. What is the probability that a student's favorite sneaker was brand D?
Sneaker
A
B
C
D
E
Number
12
15
24
26
13
Answer: There were 12 + 15 + 24 + 26 + 13 = 90 "trials" in this experiment (each student's response was a trial).
26 out of the 90 students chose brand D.
The probability is :
Theoretical Probability
With theoretical probability, you do not actually conduct an experiment. Instead, you use what you know about the situation to determine the probability of an event occurring. You may use your reasoning skills or an existing formula to arrive at your answer.
The theoretical probability of an event occurring is an "expected" probability based upon knowledge of the situation. It is the number of favorable outcomes to the number of possible outcomes.
Example: Find the probability of rolling a 6 on a fair die.
Answer: No experiment is needed. There are 6 possible outcomes when rolling a die: 1, 2, 3, 4, 5, and 6.
The only favorable outcome is rolling a 6.
The probability is :
Under the best circumstances, we would expect to roll one 6 out of every 6 rolls.
Comparing Empirical (Experimental) and Theoretical Probabilities
In an experiment, two dice are rolled 50 times and the sum of the faces are recorded in a chart, as shown at the right.
1) What is the empirical probability of rolling an 8?
2) What is the theoretical probability of rolling an 8?
3) How do the empirical and theoretical probabilities compare?
Solution: 1) Empirical (experimental) probability is the probability observed in the chart above. The 8 was rolled 8 times out of 50 rolls. The empirical probability = 8/50 = 16%.
2) Theoretical probability is based upon what is expected when rolling two dice, as seen in the "sum" table at the right. The theoretical probability of rolling an 8 is 5 times out of 36 rolls. The theoretical probability = 5/36 ≈ 13.9%.
3) The experiment rolled more 8's than would be expected theoretically.
Theoretically, you would expect to roll an 8 approximately 6.9 times.
5/36 = x/50 and x ≈ 6.9 times
Empirical (experimental) probability approaches theoretical probability
when the number of trials is extremely large.
The Law of Large Numbers (called Bernoulli's Theorem) states:
"If an experiment is repeated a large number of times, the experimental or empirical probability of a particular outcome
approaches a fixed number as the number of repetitions increases. This fixed number is the theoretical probability.
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