4.3 Finding Probability Using Sets
A compound event consists of two or more simple events.
Let A and B represent two sets of data
Intersection of Sets - 𝐴 ∩ 𝐵
Means data that are in both sets {A and B}
Union of Sets - 𝐴 ∪ 𝐵
Means data that are in either set {A or B}
Disjoint Sets - 𝐴 ∩ 𝐵 = ∅
“empty set”
Means two sets which do not have any common data
** A and B are then mutually exclusive events.
Example 1
S
A
B
Construct a Venn diagram to represent the following data for 123 grade 12 mathematics
students.
Math Course Taken
Data Management (DM)
Calculus & Vectors (CV)
College Mathematics (CM)
Data AND Calculus
Number of Students
45
27
54
3
S
DM
42
3 CV
24
CM
54
Since 𝑛(𝐷𝑀 ∩ 𝐶𝑉) = 3, then the number of students
taking just DM is 45-3=42 and just taking CV is 27-3=24
What is 𝑛(𝐷𝑀 ∪ 𝐶𝑉)?
𝑛(𝐷𝑀 ∪ 𝐶𝑉) = 𝑛(𝐷𝑀) + 𝑛(𝐶𝑉) − 𝑛(𝐷𝑀 ∩ 𝐶𝑉)
= 45 + 27 − 3
= 69 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
*Not mutually exclusive
Since these students
are counted twice
What is 𝑛(𝐷𝑀 ∩ 𝐶𝑀)?
𝑛(𝐷𝑀 ∩ 𝐶𝑀) = 0 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑐𝑖𝑛𝑐𝑒 𝐷𝑀 ∩ 𝐶𝑀 = ∅
Mutually exclusive
What is 𝑛(𝐷𝑀 ∪ 𝐶𝑀)?
𝑛(𝐷𝑀 ∪ 𝐶𝑀) = 𝑛(𝐷𝑀) + 𝑛(𝐶𝑀)
= 45 + 54
= 99 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
Additive Principle for Unions of Two Sets
If A and B are NOT mutually exclusive,
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)
This also applies for probability,
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
If A and B ARE mutually exclusive,
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵)
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
S
A
B
Determine the probability that:
i)
A student takes only the calculus & vectors course:
𝑃(𝐶𝑉) =
ii)
𝑛(𝐶𝑉)
𝑛(𝑆)
27
9
= 123 = 41
A student was in calculus & vectors OR data management:
𝑃(𝐶𝑉 ∪ 𝐷𝑀) = 𝑃(𝐶𝑉) + 𝑃(𝐷𝑀) − 𝑃(𝐶𝑉 ∩ 𝐷𝑀)
27
45
3
= 123 + 123 − 123
69
= 123
iii)
A student in calculus & vectors AND college math
𝑃(𝐶𝑉 ∩ 𝐶𝑀) =
iv)
𝑛(𝐶𝑀∩𝐶𝑉)
𝑛(𝑆)
=0
A student was in calculus & vectors OR college math
𝑃(𝐶𝑉 ∪ 𝐶𝑀) = 𝑃(𝐶𝑉) + 𝑃(𝐶𝑀)
27
54
= 123 + 123
81
= 123
Homework: page 228 #1, 3, 5, 6, 7, 9, 10, 11, 12 (read example 3 on page 226 to help with dice ques)