1. Probabilities

2. Events & Operations on it.
An Event is a set of outcomes of an experiment.
The diagram shows that A is a set of outcomes of the experiment B.

3. Events & Operations on it.
Probability P of an event A = n(A)/n(Ω).
Ω is the Sample Space.
0 ≤ P ≥ 1.
Probability of a certain event = 1.
Probability of impossible event = 0.

4. Events & Operations on it.
Intersection between two events (A∩B) means the occurrence of A and B together.
Union between two events (AUB) means the occurrence of at least one event A or B.
Rules:
P(A∩B)= (A∩B)/Ω.
P(AUB)= P(A) + P(B) - P(A∩B).
Example: A card is selected from a pack of cards numbered 1 to 19. What is the probability that the card has an odd number? What is the probability that the card has a prime number? What is the probability that the card is an odd and prime number?

5. Events & Operations on it.
Solution:
Sample space S= {1,2,3,4,5,6,7,….,19}
Odd numbers A={1,3,5,7,9,11,13,15,17,19}
Prime numbers B={1,2,3,5,7,11,13,17,19}
(A∩B)={1,3,5,7,11,13,17,19}
(AUB)={1,2,3,5,7,9,11,13,15,17,19}
So, P(A)=10/19 & P(B)=9/19
P (A∩B)= 8/19
P(AUB)= 11/19 = P(A) + P(B) - P(A∩B)
= 10/19+9/19-8/19= 11/19

6. Events & Operations on it.
Note that:
If A ⊂ B then:
(A∩B) = A and P(A∩B) = P(A).
(AUB) = B and P(AUB) = P(B).
Mutually exclusive events are the events that can’t occur together.
P(A∩B) = 0.
P(AUB) = P(A) + P(B)

7. Complementary Events
Complementary event of A is À means the occurrence of one of them only.
(AUÀ) = Ω.
P(AUÀ) = 1.
(A∩À) = Ø.
P(A∩À) = 0.
P(A) = 1 - P(À).
P(À) = 1 – P(A).

8. Difference between events
The occurrence of one event and the non occurrence of the other.
(A-B)= (A) – (A∩B).
P(A-B)= P(A) – P(A∩B).
Example: Suppose that A and B are events in a random experiment. Given that P(A)=0.53, P(B’)=0.46, and P(AUB)=0.61, determine P(B - A).

9. Difference between events
Solutions:
P(A) = 0.53
P(B) = 1 – P(B’) = 1 – 0.46 = 0.54
P(AUB) = 0.61
P(A∩B) = P(A) + P(B) – P(AUB)
= – 0.61= 0.46
P(B-A) = P(B) – P(B∩A)
= 0.54 – 0.46 = 0.08