Chapter 7-2 Theoretical and Experimental Probability
Obj: To find the Theoretical and Experimental Probability of an event
Why? - You can use probability to find the chances of hitting or missing a target in a game like Battleship (see Example 2)
Probability – is the measure of how likely an event is to occur. It is expressed as number (fraction, or decimal)
from 0 to 1 or as a percent from 0% to 100%. Zero would represent impossible, and 1 or 100% would represent
certain.
Impossible
As likely as not
0 or 0%
0.5, ½ or 50%
Certain
1 or 100%
Outcome – the result of a probability experiment or situation.
Sample Space – is the set of all possible outcomes.
Event – is an outcome or set of outcomes
Equally likely events have the same chance of occurring.
Example
Theoretical Probability – For equally likely outcomes,
𝑃(𝑒𝑣𝑒𝑛𝑡) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑓
𝑜𝑟
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑝
Ex 1. Find the Theoretical Probabilities of the following:
a. A CD has 5 upbeat dance songs and 7 slow ballads. What is the probability that a
randomly selected song is an upbeat dance song?
b. A red number cube and a blue number cube are rolled.
If all the numbers are equally likely, what is the
event?
- The sum is ten
probability of each
-The sum is 6
-The red cube is greater
The sum of all probabilities in a sample space always equals 1. If we add the probability of
an event occurring AND the probability of the event not occurring, it always equals one.
Compliment – of an event E is the set of all outcomes in the sample space that are not E
The probability of the compliment of event E is:
𝑃(𝑛𝑜𝑡 𝐸 ) = 1 − 𝑃(𝐸 )
𝑎𝑛𝑑
𝑃(𝐸 ) + 𝑃(𝑛𝑜𝑡 𝐸 ) = 1
Ex 2. There are 25 students in a study hall. The table to the
right shows the number of students who are studying a foreign
language. What is the probability that a randomly chosen
student is not studying a foreign language?
Of the students studying a foreign language, what is the probability of:
Not studying French?
Not studying Japanese?
Finding Probability with Permutations or Combinations
Ex 3. Each student received a 4 digit code to use the library computers, with no digit
repeated. What is the probability that someone would receive a code of consecutive
numbers?
1st – is it a permutation or combination?
2nd – Find the number of outcomes in the sample space
3rd – find the number of favorable outcomes (1234, 2345,…AND 9876, 8765…)
4th – Find the probability 𝑃(𝐸) =
𝑓
𝑝
= ___________
Geometric Probability – is a form of theoretical probability determined by a ratio of
lengths, areas, or volumes.
Ex.4 Find the probability that a point chosen at random inside the
large triangle is in the small triangle
Experimental Probability – of an event is the ratio of the number of times that the event
occurs (frequency) to the number of trials. (basically…its what you actually get when you
do the trials)
𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Ex. 5 The table shows the results of a spinner
experiment. Find each experimental probability.
-Spinning a 4
-Spinning a number greater than 4
7.2 – pg 494, 2–9, 12-16, 19, 20