Basics of probability

A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1. A probability of 0 indicates that there is no chance that a particular event will occur, whereas a probability of 1 indicates that an event is certain to occur.

Probability of an Event:

$$ P(A) = \frac {n(E)}{n(S)} $$

Where,

n(E) = Number of favorable outcomes

n(S) = Total number of possible outcomes

Example:

Probability of obtaining an odd number on rolling dice for once.

Solution:

Sample Space = {1, 2, 3, 4, 5, 6}

n(S) = 6

Favorable outcomes = {1, 3, 5}

n(E) = 3

Using the probability formula,

$$ P(A) = \frac {n(E)}{n(S)} $$

$$ \text{P(Getting an odd number)} = \frac 36 $$

$$ = \frac 12 = 0.5 $$

Probability Range:

$$ 0 \le P(A) \le 1 $$

Rule of Complementary Events:

$$ P(A’) = 1- P(A) $$

Example:

Find the probability that a leap year has 52 Sundays.

Solution:

A leap year can have 52 Sundays or 53 Sundays.

In a leap year, there are 366 days out of which there are 52 complete weeks & remaining 2 days.

Now, these two days can be (Sat, Sun), (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat).

So there are total 7 cases out of which (Sat, Sun) (Sun, Mon) are two favourable cases.

So, P (53 Sundays) \( = \frac 27 \)

So, P (52 Sundays) = 1 – P(53 Sundays)

\( = 1 – \frac 27 \)

\(= \frac 57 \)

Rule of Addition:

$$ P(A\cup B) = P(A) + P(B) - P(A\cap B) $$

Disjoint Events

Events A and B are disjoint if

$$ P(A\cap B) = 0 $$

Independent Events:

$$ P(A) = P(A|B) $$

$$ P(B) = P(B|A) $$

$$ P(A\cup B) = P(A) + P(B) $$

$$ P(A\cap B) = P(A) \cdot P(B) $$

Example:

A single fair die is rolled. Let A={3} and B={1,3,5} Are A and B independent?

Solution:

In this example we can compute all three probabilities

\( P(A)= \frac 16 \)

\( P(B)= \frac 12 \)

\( P(A\cap B)=P({3})= \frac 16 \)

\( P(A)\cdot P(B)=(\frac 16)(\frac 12)=\frac {1}{12} \)

\( P(A\cap B) \neq P(A) \cdot P(B) \)

So,the events A and B are not independent

Conditional Probability:

$$ P(A | B) = P(A\cap B) / P(B) $$

Bayes Formula:

$$ P(A | B) = \frac {P(B | A) \cdot P(A)}{P(B)} $$