# Theoretical vs. experimental probability

### Theoretical vs. experimental probability

In this lesson, we will learn:

• The difference between theoretical and experimental probability
• How to calculate the number of expected outcomes using theoretical probability and number of experimental trials
• How to write the experimental probability as a fraction based off the observed results in an experiment

Notes:

• Probability for simple events means we are just looking at one probability scenario at a time (i.e. one coin flip; a single six-sided die toss; one spinner)

• There are two types of probability:
• Theoretical probability is an educated guess on what you think should or will happen ("expected" probability; based on theory)
• Experimental probability is based on an experiment and what you saw happen already ("observed" probability; happened in reality)

• The probability we have seen so far in previous lessons is based on theoretical probability. We can use theoretical probability to find the number of expected outcomes.
• Before you do an experiment, you can predict how many times an outcome will happen (how many times it should theoretically happen)

$P$ (event) = $\frac{number\,outcomes\,wanted} {total\,number\,possible\,outcomes}$

• This is based on the number of trials you will do in the experiment. A trial is each run through of the experiment--1 trial will give 1 outcome (each coin flip, each dice toss, each spinner spin)

# expected outcomes = $P$ (event) × # trials

• The experimental probability is based off the observations made during the experiment and calculated once all trials are completed.

$P$ (experimental event) = $\frac{number\,outcomes\,observed} {total\,number\,trials}$

Ex. For an experiment, a coin is flipped 8 times. The results are shown in an observation table:
• The experimental (exp.) probabilities do not match with the theoretical (theor.) probabilities.
• The more trials you do, the closer the results should get to the expected probabilities (i.e. doing 10 trials vs. 100 trials vs. 1000 trials, etc.)

#### Lessons

• Introduction
Introduction to Basic Probability for Simple Events:
a)
Theoretical probability and expected outcomes

b)
Finding experimental probability and comparing with experimental probability

c)
Giving experimental probability as a fraction

d)
Review on theoretical probability vs. experimental probability

• 1.
Theoretical probability and expectations
a)
How many times would you expect to land on heads if you flipped a coin 10 times?

b)
How many times would you expect to roll the number 2 if you toss a six-sided die 30 times?

c)
How many times would you expect to land on the letter A if you spin a four-part spinner 40 times?

d)
The experiment will consist of pulling 1 lollipop out of the bag at a time. Each lollipop is put back into the bag before the next pull.

1. How many times would you expect to pull a red lollipop if you tried 20 times?
2. How many times would you expect to pull a red lollipop if you tried 20 times?

e)
How many times would you expect to land on a number greater than 4 if you toss a six-sided die 9 times?

• 2.
Comparing experimental probability and theoretical probability (results in reality vs. expectation)
A student is preparing to conduct a probability experiment flipping a coin. They will record their results in a table:

a)
What (theoretical) probability fraction is expected for each outcome?

b)
Which of the following seems most likely to happen if the coin will be flipped 100 times?