CBSE Class 12 Mathematics Probability Notes Set 02

BASIC CONCEPTS
1. Probability: Probability is a branch of mathematics in which the chance of an event happening is assigned a numerical value that predicts how likely that event is to occur.

Random Experiment: The experiment, in which the outcomes may not be same even if the experiment is performed in identical condition, is called random experiment. e.g., Tossing a coin is a random experiment because if we toss a coin in identical condition, outcomes may be head or tail.

2. Outcome: An outcome is a result of some activity or experiment.

3. Sample Space: A sample space is a set of all possible outcomes for a random experiment.

4. Event: An event is a subset of the sample space.

5. Theoretical Probability: The theoretical probability of an event is the number of ways that the event can occur, divided by the total number of possibilities in the sample space.
Symbolically, we write \( P(E) = \frac{n(E)}{n(S)} \), where \( P(E) \) represents the probability of the event.

6. In general, for any sample space \( S \) containing \( k \) possible outcomes, we say \( n(S) = k \). When the event \( E \) is certain, every possible outcome for the sample space is also an outcome for event \( E \) or \( n(E) = k \). Thus, the probability of a certain or sure event is given as
\[ P(E) = \frac{n(E)}{n(S)} = \frac{k}{k} = 1 \]
Note:
(i) The probability of an event that is certain to occur is 1.
(ii) The probability of any event \( E \) must be equal to or greater than 0; and less than or equal to 1, i.e., \( 0 \le P(E) \le 1 \).

7. Another way of expressing probability is in term of axioms, laid by Russian mathematician A. N. Kolmogorov.
If \( S \) is a sample space, then probability \( P \) is a real valued function defined on \( S \) and take values \( [0, 1] \), satisfying following axiom
(i) Probability of any event \( \ge 0 \).
(ii) Sum of probabilities assigned to all members of \( S \) is 1.
(iii) For any two mutually exclusive events \( E \) and \( F \), \( P(E \cup F) = P(E) + P(F) \).

8. Theorems of Probability:
(i) Addition theorem:
(a) When the events are not mutually exclusive: The probability that at least one of the two events \( A \) and \( B \) which are not mutually exclusive will occur is given
Symbolically, \( \quad P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
In the case of three events:
\( P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C) \)
(b) When \( A \) and \( B \) are mutually exclusive: The addition theorem states that if two events \( A \) and \( B \) are mutually exclusive, the probability of the occurrence of either \( A \) or \( B \) is the sum of the individual probability of \( A \) and \( B \)
Symbolically, \( \quad P(A \cup B) = P(A) + P(B) \)
The theorem can be extended to three or more mutually exclusive events
thus, \( \quad P(A \cup B \cup C) = P(A) + P(B) + P(C) \)
(ii) Multiplication theorem: This theorem states that if two events \( A \) and \( B \) are independent, the probability that they both will occur is equal to the product of their individual probabilities.
Symbolically, \( \quad P(A \cap B) = P(A) . P(B) \)
The theorem can be extended to three or more independent events
thus, \( \quad P(A \cap B \cap C) = P(A) . P(B) . P(C) \)
Note: If \( A \) and \( B \) are mutually exclusive and exhaustive, then
\( P(A \cup B) = P(A) + P(B) = 1 \)

9. A rule for the probability of the event not A: If \( P(A) \) is the probability that some given event will occur, and \( P(\text{not } A) \) is the probability that the given event will not occur, then
Symbolically: \( \quad P(A) + P(\text{not } A) = 1 \) or \( P(A) = 1 - P(\text{not } A) \) or \( P(\text{not } A) = 1 - P(A) \)
We write \( P(\text{not } A) \) as \( P(\bar{A}) \).

10. Problems, related to withdrawal of balls, cards, letters, etc. with replacement and without replacement:
In such type of problems, the sample space will not change when the articles (balls, cards, letters, etc.) are replaced after each withdrawal. While in case when the article is not replaced (without replacement), the sample space will change after each withdrawal.
Note:
(i) If the problem does not specifically mention "with replacement" or "without replacement", ask yourself: "Is this problem with or without replacement?"
(ii) For many compound events, the probability can be determined most easily by using the counting principle i.e., permutations and combinations.
(iii) Every probability problem can always be solved by

• Counting the number of elements in the sample space \( n(S) \);
• Counting the number of outcomes in the events, \( n(E) \);
• And substituting these numbers in the probability formula.
  \[ P(E) = \frac{n(E)}{n(S)} \]

(iv) Taking out 2 or more objects (e.g. balls) randomly from a bag one by one without replacement is same as taking out 2 or more objects is same as taking out 2, or more objects simultaneously.
The number of ways in which \( r \) objects can be taken out of \( n \) objects is \( ^nC_r \) or \( C(n, r) = \frac{n!}{(n - r)! . r!} \).

11. Conditional Probability: If \( A \) and \( B \) are two events associated with the some random experiment, then the probability of occurrence of event \( A \), when the event \( B \) has already occurred is called conditional probability of \( A \) when \( B \) is given. It is represented by \( P(A/B) \) and is given by
\( P\left(\frac{A}{B}\right) \) = Probability of event \( A \) when \( B \) has already occurred
= Probability of event '\( A \cap B \)' when \( B \) behaves like sample space
= \( \frac{n(A \cap B)}{n(B)} \)
= \( \frac{\frac{n(A \cap B)}{n(S)}}{\frac{n(B)}{n(S)}} \) [Dividing \( N' \) and \( D' \) by \( n(S) \)]
= \( \frac{P(A \cap B)}{P(B)} \)
Similarly, \( P\left(\frac{A}{B}\right) = \frac{P(A \cap B)}{P(A)} \)

Theorem of Total Probability: Let \( E_1, E_2, ..., E_n \) be the events of a sample space '\( S \)' such that they are pair wise disjoint, exhaustive and have non-zero probabilities. If \( A \) is any event associated with \( S \), then
\[ P(A) = P(E_1).P\left(\frac{A}{E_1}\right) + P(E_2).P\left(\frac{A}{E_2}\right) + .... + P(E_n).P\left(\frac{A}{E_n}\right) \]

12. Baye's Theorem: If \( B_1, B_2, ... B_n \) are mutually exclusive and exhaustive events and \( A \) is any event that occurs with \( B_1 \) or \( B_2 \) or \( B_n \) then
\[ P(B_i / A) = \frac{P(B_i).P(A / B_i)}{\sum\limits_{i=1}^n P(B_i).P(A / B_i)}, i = 1, 2, ..., n. \]
Note: The probabilities \( P(B_i) \) \( i = 1, 2, ..., n \) which were already known before performing an experiment are known as prior probabilities and conditional probabilities \( P(B_i / A) \), \( i = 1, 2, 3, ..., n \) which are calculated after the experiment is performed are known as posterior probabilities. The events \( B_1, B_2, ... B_n \) are usually called causes for event \( A \) to occur.

13. Random Variable: Random variable is simply a variable whose values are determined by the outcomes of a random experiment; generally it is denoted by capital letters such as \( X, Y, Z \), etc. and their values are denoted by the corresponding small letters \( x, y, z \), etc.

14. Probability Distribution: The system consisting of a random variable \( X \) along with \( P(X) \) is called the probability distribution of \( X \).

15. Mean and Variance of a Random Variable: Let a random variable \( X \) assume values \( x_1, x_2, .... x_n \) with probabilities \( p_1, p_2, ... p_n \) respectively, such that \( p_i \ge 0, \sum\limits_{i=1}^n p_i = 1 \). Then, the mean of \( X \), denoted by \( \mu \), [or expected value of \( X \) denoted by \( E(X) \)] is defined as
\[ \mu = E(X) = \sum\limits_{i=1}^n x_i p_i \] and
Variance denoted by \( \sigma^2 \) is defined as
\[ \sigma^2 = \sum\limits_{i=1}^n (x_i - \mu)^2 p_i = \sum\limits_{i=1}^n x_i^2 p_i - \mu^2 \]

16. Standard Deviation, \( \quad \sigma = \sqrt{\text{variance}} \)