Probability
This chapter deals with probability, the concept of probability is also studied in earlier classes. This chapter in the present class helps to learn about conditional probability. Further, topics like Bayes’ theorem, independence of events, the probability distribution of random variables, mean and variance of a probability distribution, and Binomial distribution are discussed in this chapter.
Conditional Probability: The possibility of an event or outcome occurring dependent on the occurrence of a preceding event or outcome is known as conditional probability. It simply depends on any previous occurrence that has already occurred. Consider two events A and B with the same sample space of a random experiment, then the conditional probability of the event A gives that B has occurred, i.e. P(A|B) is,
P(A|B) = P(A ∩ B)/P(B)
(for P(B) ≠ 0).
- When E and F are the events of a sample space S of an experiment: P(S|F) = P(F|F) = 1
- When A and B are any two events in a sample space S and F an event of S, such that P(F)≠0: P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)
- P(E′|F) = 1 − P(E|F)
Multiplication Rule: Consider two events such as, E and F from a sample space S. Here, the set E ∩ F denotes the event that both E and F have occurred. Or we can say, E ∩ F represents the simultaneous occurrence of the events E and F. The event E ∩ F is also written as EF. According to this rule, if E and F are the events in a sample space, then;
P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F)
where P(E) ≠ 0 and P(F) ≠ 0.
Similarly, for three events E, F, and G from a sample S:
P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)
Independent Events: Two experiments are said to be independent if the probability of the events E and F occurring simultaneously when the two experiments are performed is the product of P(E) and P(F) calculated separately on the basis of two experiments, i.e., for every pair of events E and F, where E is associated with the first experiment and F with the second experiment.
P (E ∩ F) = P (E).P(F)
Baye’s Theorem: A set of events E1, E2, …, En is said to denote a partition of the sample space S when,
- Ei ∩ Ej = φ, i ≠ j, i, j = 1, 2, 3, …, n
- E1 ∪ Ε2 ∪ … ∪ En = S and
- P(Ei )> 0 for all i = 1, 2, …, n.
Also, the events E1, E2, …, En denotes a partition of the sample space S if they are pairwise disjoint, exhaustive, and have nonzero probabilities, and A be any event with non-zero probability, so:
[Tex]P(E_i|A)=\dfrac{P(E_i)P(A|E_i)}{\sum_{j=1}^{n}P(E_j)P(A|E_j)} [/Tex]
Theorem of Total Probability: Let E1, E2, …., En be the partition of a sample space and A be any event; then,
P(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + … + P (En) . P(A|En)
Random Variables and their Probability Distributions: A random variable is a real-valued function whose domain is a random experiment’s sample space. The probability distribution of a random variable X is the system of numbers:
X: x1 x2 … xn and P(X): p1 p2 … pn where pi > 0,
[Tex]\sum_{i=1}^{n}p_i=1 [/Tex]
where i = 1, 2, 3, … , n.
The real numbers x1, x2, …, xn are the possible values of the random variable X, and pi (i = 1, 2, …, n) is the probability of the random variable X taking the value xi i.e.
P(X = xi ) = pi
The mean, variance and standard deviation of a random variable X can be written as:
- Mean: [Tex]E(x)=\mu=\sum^{n}_{i+1}x_ip_i [/Tex]
- Variance: [Tex]\sigma^2_x=Var(X)=\sum^{n}_{i=1}(x_i-\mu)^2p(x_i)=E(X-\mu)^2 [/Tex]
- Standard Deviation: [Tex]\sigma_x=\sqrt{Var(X)}=\sqrt{\sum^{n}_{i=1}(x_i-\mu)^2p(x_i)} [/Tex]
Bernoulli Trials and Binomial Distribution: Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions:
- There should be a finite number of trials.
- The trials should be independent.
- Each trial has exactly two outcomes: success or failure.
- The probability of success remains the same in each trial.
P (X = x) = P(x) = nCx qn-x px
where,
- x = 0, 1, …, n,
- n is the Total number of trials
- p is the probability of success
- q = 1 – p
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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