Hey Developer’s, I’m back with a new topic which is Rules of Probability in the series of statistics foundations.

### Quick Refresher

**Probability : It is how likely an event to occur**

So, let’s get started …

### Rules of Probability

- Probability of event A is greater than or equal to zero : P(A) >= 0
- Probability of sample space is one: P(S) = 1
- The sum of probabilities of all possible events equals 1

**Rule of Subtraction**: P(A) = 1 – P(A’).

Example: Suppose, for example, the probability that Bill will graduate from college is 0.80. What is the probability that Bill will not graduate from college? Based on the rule of subtraction, the probability that Bill will not graduate is 1.00 – 0.80 or 0.20.

**Rule of Multiplication** : P(A ∩ B) = P(A) P(B|A)

The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

Example : An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn *without replacement* from the urn. What is the probability that both of the marbles are black?

*Solution:* Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:

- In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
- After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

Therefore, based on the rule of multiplication:P(A ∩ B) = P(A) P(B|A)

P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133

**Rule of Addition** : P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

The probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur.

Note: Invoking the fact that P(A ∩ B) = P( A )P( B | A ), the Addition Rule can also be expressed as:

P(A ∪ B) = P(A) + P(B) – P(A)P( B | A )

Example: A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?

*Solution:* Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:P(F ∪ N) = P(F) + P(N) – P(F ∩ N)

P(F ∪ N) = 0.40 + 0.30 – 0.20 = 0.50

#### Next Post will be on Discrete and Continuous Probability

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