Probability and Probability Distributions - Part2
Probability of Multiple Events
Rule of addition for mutually exclusive events (disjoint events)
Consider two mutually exclusive events A and B then P(A or B) = P(A) + P(B)
Rule of Addition for Not Mutually Exclusive Events
Consider two events, A and B, not mutually exclusive then P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) is subtracted to avoid double counting.
Rule of Multiplication for Independent Events
Two events, A and B, are considered as independent if the occurrence of A does not affect in any way the occurrence of B.
Then the joint probability of A and B is given by: P(A and B)= P(A) × P(B)
Rule of Multiplication for Dependent Events
Two events are dependent if the occurrence of one affects somehow the occurrence of the other. Then P(A and B)= P(A) − P(B/A)
Rule of Multiplication for Dependent Events
We read this as "The probability that both events A and B will take place equals the probability of event A times
the probability of event B, given that event A has already occurred."
P(B/A) = conditional probability of B, given that A has already occurred and P(A and B) = P(B and A)
Examples
Example 5
Consider a single toss of a die, we can get only one of six possible outcomes: I, 2, 3, 4, 5, or 6, which are mutually exclusive events.
If the die is fair then: P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6.
The probability of getting a 4 or a 5 on a single toss of the die is
P(4 or 5) = P(4) + P(5) = 1/6 + 1/6 = 2/6 = 1/3
Similarly, P(4 or 5 or 6) = P(2) + P(3) + P(4) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
Example 6
The outcomes of two successive tosses of a fair coin are independent events.
The outcome of the first toss does not affect the outcome on the second one.
Hence the following: P(H and H) = P(H ∩ H) = P(H) · P(H) = 1/2 + 1/2 = 1/4 = 0.25
Similarly, P(T and T and T) = P(T ∩ T ∩ T ) = P(T) · P(T) · P(T) = 1/2 × 1/2 × 1/2 = 1/8 = 0.125
For more details, please contact me here.
Date of last modification: March 25, 2019