Revision: Probability >> Probability Maths Commerce (English Medium) Class 12 CBSE

The conditional probability of both events A and B over the sample space S is

Two events are said to be independent if the occurrence of one does not depend on the other.

For two events E and F:

• E and F are independent if P(F | E) = P(F), when \[P(E) \neq 0\].

• Similarly, E and F are independent if P(E | F) = P(E), when \[P(F) \neq 0\].

• An equivalent and most commonly used test is:

Statement: 

Let p be the probability of success of an event and q be the probability of failure of the event in one trial. Suppose there are n trials of the event in a binomial  experiment, then the binomial probability distribution is defined by the following table: 

Trials of a random experiment are called Bernoulli’s trials if they satisfy the following conditions:

1. The number of trials is finite.

2. Each trial is independent of the others.

3. Each trial has exactly two outcomes: success or failure.

4. The probability of success (or failure) remains the same in each trial.

Independent events:

 A set of events is said to be independent if the occurrence of any one of them does not, in any way, affect the occurrence of any other in the set.

Dependent events:

 Two events E and F are said to be dependent if they are not independent, i.e. if \[\mathrm{P}(\mathrm{E}\cap\mathrm{F})\neq\mathrm{P}(\mathrm{E}).\mathrm{P}(\mathrm{F})\]

Random variable:

A random variable is a variable whose values depend on chance and are the result of a random observation or experiment.

Discrete random variable:

If the set of values taken by a random variable can be counted and listed, it is called a discrete random variable.

Continuous Random Variable:

If the set of values is continuous, the variable is called a continuous random variable.

If a random variable x can take values x₁, x₂,…, x_n with probabilities p(x₁) ,p(x₂),…, p(x_n) such that p(x₁) + p(x₂) +… +  p(x_n) = 1, the function p is called the probability density function of x and is said to define the probability distribution of x.

Mean µ (Greek mu) of the above probability distribution may be defined as 

\[\mu=\frac{p_1x_1+p_2x_2+p_3x_3+.......+p_nx_n}{p_1+p_2+p_3+......+p_n}\]

\[=\frac{\sum p_ix_i}{\sum p_i}=\Sigma p_ix_i\]

\[Mean\overline{x}=\sum_{i=1}^{n}p_{i}x_{i}\],where each pi \[P_{i}\geq0\] and \[\sum p_{i}=p_{1}+p_{2}+...+p_{n}=1\]

\[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].

\[P(B | A) = \frac{P(A \cap B)}{P(A)}\], where \[P(A) \neq 0\].

Multiplication rule: \[P(A \cap B) = P(B) \cdot P(A | B) = P(A) \cdot P(B | A)\].

Complement form: \[P(A' | B) = 1 - P(A | B)\].

\[\begin{gathered}
\mu=\int_{-\infty}^{\infty}xf(x)dx \\
\sigma^2=\int_{-\infty}^\infty(x-\mu)^2f(x)dx
\end{gathered}\]

\[\sigma=\sqrt{\sigma^2}=\sqrt{\sum p_ix_i^2-\mu^2}\]

General Form: \[P(X=r)={}^nC_rp^rq^{n-r},\quad r=0,1,2,\ldots,n\]

Mean: μ = np

Variance: σ² = npq

Standard deviation: \[\sigma=\sqrt{npq}\]

The variance of a random variable x is denoted by σ².

First form: \[\sigma^2=\sum_{i=1}^np_i(x_i-\mu)^2\]

Second form: \[\sigma^2=\sum_{i=1}^np_ix_i^2-\mu^2\]

For two events:

• \[P(E \cap F) = P(F) \cdot P(E | F)\]

• \[P(E \cap F) = P(E) \cdot P(F | E)\]

For three events:

If B₁, B₂,..., Bn are mutually exclusive and exhaustive events and if A is an event consequent to these B_i's, then for each i = 1, 2, 3, ..., n,

Statement: 
Let S be the sample space and E₁, E₂,…, E_n be mutually exclusive and exhaustive events associated with a random experiment. Let A be any event associated with S. Then,

\[P(A)=P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)+\cdots+P(E_n)P(A\mid E_n)\]

or

\[P(A)=\sum P(E_i)P(A\mid E_i)\]

• Conditional probability means probability under a given condition.

• The formula is \[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].

• Always reduce the sample space according to the condition first.

• The numerator represents outcomes common to both events.

• Do not confuse P(A | B) with P(B | A).

• For independent events, P(A | B) = P(A).

• Multiplication theorem is used to find the probability of simultaneous occurrence of events.

• For two events: \[P(E \cap F) = P(E) \cdot P(F | E)\]

• Another equivalent form is \[P(E \cap F) = P(F) \cdot P(E | F)\].

• For three events: \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\].

• Most “without replacement” questions are solved using this theorem.

• Always define events before solving a probability problem.

• Independent events do not influence each other.

• The standard test is \[P(E \cap F) = P(E)P(F)\].

• Conditional form: \[P(F | E) = P(F)\] and \[P(E | F) = P(E)\], when defined.

• If two events are independent, related complement pairs are also independent.

• Mutually exclusive events and independent events are different.

• For independent events A and B, \[P(A \cup B) = 1 - P(A')P(B')\].

• For three events, mutual independence requires pairwise conditions and the condition involving all three together.

• Bayes' Theorem works from effect to cause.

• Prior probability means “before observation”.

• Posterior probability means “after observation”.

• The observed event is usually given in the question statement.

• The required answer is often a probability of the form P(cause ∣ observed event).

• Probabilities are terms of (q + p)ⁿ.

• P(0) + P(1) + ⋯ + P(n) = 1.

• The binomial distribution is discrete.

• n and p are its parameters.

Special cases:

• P(0) = qⁿ

• P(1) = npqⁿ⁻¹