In this post, let’s learn the basics of Probabilities.
Probability
For a random experiment with n elementary outcomes: { O1, O2, O3, … , On}, we can assign a probability to each outcome.
The probability of the outcome Oi is written as Pr(Oi) or P(Oi).
- P(Oi) ≥ 0
- The probability is non-negative.
- P(O1) + P(O2) + … + P(On) = 1
- Total probability of all elementary outputs is ONE.
[Example] the throw of a single dice
- P(1) = 1/6
- P(2) = 1/6
- P(3) = 1/6
- P(4) = 1/6
- P(5) = 1/6
- P(6) = 1/6
Basic Operations
Given Events A and B, we can make new events.
- P (A and B) : Both A and B occur.
- P (A or B) : A or B (or both of them) occurs
- P (not A) : A does not occur
- Addition Rule
- P (A or B) = P(A) + P(B) – P(A and B)
- Subtraction Rule
- P(not A) = 1 – P(A)
- P(A) = 1 – P(not A)
[Example] the throw of two dices (black and white)
- Event A : white dice is 1
- Event B: black dice is 1
- P(A) = 6/36 = 1/6
- P(B) = 6/36 = 1/6
- P(A and B) = 1/36
- P(A or B) = P(A) + P(B) – P(A and B) = 6/36 + 6/36 – 1/36 = 11/36
- P(not A) = 1 – P(A) = 1 – 1/6 = 5/6
Conditional Probability
The probability of an event A, given the condition that the event B already occurred, is written like this:
- P(A | B) reads “the probability of A given B”
[Example] the throw of two dices (black and white)
- Event A: the sum of the 2 dices are 3
- P(A) = 2/36 = 1/18
- { {1,2}, {2,1}}
- P(A) = 2/36 = 1/18
- The probability of an Event A when the white dice is already thrown and the value is one (Event B)
- P(A|B) = 1/6
- intuitively, the white is already 1 and the black should be 2
- P(B) = 1/6
- P(A and B) = 1/36
- P(A|B) = P(A and B) / P(B) = (1/36) / (1/6) = 6/ 36 = 1/6
Multiplication Rules
P(A and B) = P(A│B)×P(B) = P(B│A)×P(A)
- Special Case: when A and B are independent
- P(A | B) = P(A)
- P(A and B)= P(A) × P(B)
[Example] the throw of two dices (black and white)
- Event A : white dice is 1
- Event B: black dice is 1
- Event C : the sum is 3
- Event D: white dice is 1
- P (A | B) = P (A and B) / P(B) = (1/36) / (1/6) = 1/6
- P (A | B) = P(A) = P(B)
- P(A and B) = P(A) * P(B) = (1/6) * (1/6) = 1/36
- P (C | D) = P (C and D) / P(D) = (1/36) / (1/6) = 1/6
- P(C) = 2/36 = 1/18
- P(C|D) ≠ P(C)
- P(C and D) = P(C|D) * P(D) = (1/6) * (1/6) = 1/36
Scriptorium 