The Plan
1. Basic Probability Rules
2. DeMorgan's Law
3. Mutually Exclusive Events
4. Independent Events
5. Mutually Exclusive vs. Independent Events
6. Example Questions
1. Basic Probability Rules
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴𝐵)
𝑃(𝐴𝐵) = 𝑃(𝐴)𝑃(𝐵 |𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵)
𝑃(𝐴̅)−= 1 − 𝑃(𝐴)
2. DeMorgan's Law
𝐴 ∪ 𝐵 = 𝐴̅ ∩ 𝐵 and 𝐴 ∩ 𝐵 = 𝐴̅ ∪ 𝐵
Hence 𝑃(𝐴 ∪ 𝐵 ) = 𝑃(𝐴̅ ∩ 𝐵) and 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴̅ ∪ 𝐵)
3. Mutually Exclusive Events
Two events are mutually exclusive (or disjoint) iff 𝑃(𝐴𝐵 ) = 0.
4. Independent Events
Two events are independent iff P(A|B) = P(A) iff P(B|A) = P(B)
One might also say that
two events are dependent iff P(A|B) is not equal to P(A) iff P(B|A)=P(B).
5. Mutually Exclusive vs. Independent Events
Mutually Exclusive
Not Mutually
Exclusive
Independent
Dependent
6. Examples Questions
a. Draw Venn diagrams for each event.
i.
𝐴∪𝐵
ii.
𝐴̅ ∪ 𝐵
iii.
𝐴∩𝐵
iv.
(𝐴 ∩ 𝐵) ∪ 𝐶̅
v.
(𝐴̅ ∪ 𝐵)
b. Give two ways to find the probability of each event in part a.
c. A fair 20 sided die is rolled three times and the numbers
observed are recorded.
i.
What is the probability that exactly one of the numbers
observed is even.
ii.
What is the probability that at least one of the numbers
observed is even.
iii.
What is the probability that at most one of the numbers
observed is even?
d. The balls are drawn, without replacement, from an urn
containing 5 red balls and 5 blue balls.
i.
What is the probability that exactly one red ball is
chosen?
ii.
What is the probability at least one red ball is chosen?
iii.
What is the probability at most one red ball is chosen?