CBSE Class 12 Mathematics Notes Chapter 13 Probability

Probability refers to the occurrence of an event based on the number of outcomes. It is indicated by 0 or 1; the larger the probability, the more likely an event is to occur.

• The probability of an event is determined by dividing the required favorable outcomes by the total number of outcomes.
• Outcome is defined as the result of an event.
• The value of the number of outcomes cannot be equal to negative.
• Sample space refers to the collection of all the outcomes for a particular event.
• Tossing a coin or selecting a card from a deck of cards are the most common examples of probability.
• The concept of probability is used in the field of probability distribution.
• A probability distribution is a statistical function that describes the likelihood that a random variable can take place within a given range.

According to the CBSE Syllabus 2023-24, the chapter on Probability comes under Unit 6. NCERT Class 12 Mathematics Unit Probability holds a weightage of around eight marks.

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Multiplication Theorem on Probability

• Multiplication theorem on probability is used to explain the conditional probability that event A will occur given that event B occurs.
• It states that the probability of occurrence of both events A and B is equal to the product of the probability of event B.
• Consider two dependent events, A and B, then the probability of both events occurring simultaneously is given by P(A ∩ B) = P(B) . P(A|B).
• Similarly, if there are two independent events, A and B, then the probability of both events occurring simultaneously is given by P(A ∩ B) = P(A) . P(B).

Multiplication Theorem on Probability

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Independent Events

• Independent events are a type of event that is independent of any other event.
• In simpler terms, if the probability of occurrence of X is not affected by the probability of occurrence of Y, then X and Y are said to be independent events.
• These events have common outcomes but are independent of maiden trials. 
• If A and B are two independent events, then the probability of occurrence of at least one event is given by 1 - P(A’)P(B’).

Independent Events

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Total Probability

• Total probability is the fundamental theorem of probability that specifies the relationship between the probability of a composite event and the probability of its sub-events. 
• It consists of a finite or countable number of mutually exclusive events.
• The total probability theorem states that the probability of occurrence of an event B is the sum of probabilities of individual events.
• The concept is used in the field of conditional probability and Bayes's theorem.
• It is also known as the Law of Total Probability.
• Suppose there are two events, X and Y, in a sample space S; then these spaces can be represented as X ∩ Y′, X ∩ Y, X′ ∩ Y, and X′ ∩ Y′.
• It can be represented as: \(P(X) = P(Y_1) + P(Y_2) + P(Y_3) + ... + P(Y_n)\), where X and Y are two events in the sample space.

Total Probability

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Conditional Probability

• Conditional probability is one of the most important concepts used in statistics.
• It refers to the occurrence of an event in conjunction with one or more other events.
• In other words, conditional probability refers to the probability of occurrence of an event X when another event Y in relation to X has already occurred. 
• It is denoted by P(X|Y).
• Mathematically, it can be represented as:

P(X|Y) = N(X∩Y)/N(Y)

• Where P(X|Y) represents the probability of occurrence of A given B has occurred.
• N(X ∩ Y) is the number of elements common to both X and Y.
• N(Y) is the number of elements in Y which cannot be equal to zero.

Conditional Probability

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Bayes Theorem

• Bayes theorem specifies occurrence of an event based on the condition that has already happened.

• It is a special case of conditional probability.
• Both the events in the theorem are mutually exclusive to each other.
• Suppose \(X_1\), \(X_2\),…, \(X_n\) are a set of events of a sample space Y, where all the events X,, X2,…, Xn indicate nonzero probability of occurrence. 
• Let B be any event or a condition associated with sample Y, then according to Bayes theorem,

P(Xi|B) = P(Xi)P(B|Xi) / ∑P(Xk)P(B|Ek)

• where, P(Xi) is the probability of event Xi. 
• P(Xi|B) is the probability of event Ei given that event B has already occurred.
• P(B|Xi) is the probability of event B given that event Xi has already occurred.

Bayes Theorem

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Random Variables

• Random variables are real valued functions that are used to represent the uncertainty of an event.
• It is used to assign a numerical value to each of the outcomes in the sample space.
• The variables are also known as random quantity, aleatory variable, or stochastic variable.
• It is used in the fields of graph theory, natural language processing and machine learning.
• Continuous Probability Distribution and Discrete Probability Distribution are two types of random variables.
• The concept is used in the mapping of sample spaces to real numbers.

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Continuous Probability Distribution

• Continuous Probability Distribution, also known as Cumulative Probability Distribution, is a type of random variable where the value of the function is equal to the variable.
• It can take on an infinite number of values.
• Continuous Probability Distribution can be represented as:

\(F_X(x)=P(X \leq x)\)

• where Fx(x) is the function of X 
• P is the probability where the value of X is less than or equal to x.

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Discrete Probability Distribution

• A discrete probability distribution is a type of variable that assumes only specified values in an interval.
• It can take on a finite or countable number of events.
• The value of probability for these variables lies in the range of 0 to 1.
• It can be represented as:

P(x)=n!r!(n−r)!⋅pr(1−p)n−r and P(x)=C(n,r)⋅pr(1−p)n−r

• Where x represents random variables.

Discrete Probability Distribution

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Mean of Random Variable

• Mean of a random variable refers to the weighted average of all the possible values that any random variable can have.
• The different probability distributions can have the same set of values.
• It cannot explain the variability of values in probability distribution. 
• Suppose Y is the random variable and A is the respective probabilities then the mean of a random variable is given by the formula:

Mean of Random Variable = ∑ YA

• where Y is set of all possible values of variables
• A is the set of respective probabilities.

\(\mu_x=x_1p_1+x_2p_2+...x_kp_k=\sum x_ip_i\)

Mean of Random Variable

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Binomial Probability Distribution

• A binomial probability distribution is a discrete probability distribution that returns only two possible values of an outcome.
• The two possible outcomes represent the rate of success or the rate of failure.
• The parameters n and p represent the number of successes in a sequence of n independent events, each with its boolean-valued outcome(p).
• Boolean-valued outcomes represent either the value of success(probability p) or failure (probability (1 − p)).
• It is also known as the Bernoulli experiment.
• The binomial probability distribution formula is given as:

\(P(x:n,p) =\; ^nCx \;p^x (1-p)^{n-x}\)

Or

\(P(x:n,p) =\; ^nCx p^x (q)^{n-x}\)

Binomial Distribution

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Poisson Probability Distribution

• Poisson Probability Distribution is a probability distribution method that returns the probability of an event taking place n number of times within the required interval of time.
• λ (lambda) is the parameter of the distribution that returns the mean of the number of events.
• It is the limited case of binomial distribution.
• In these types of distribution, the probability of one event doesn’t affect the probability of occurrence of another event.
• The Poisson distribution formula is given as:

\(f(x) =\frac{(e^{– λ} λ^x)}{x!}\)

• Where,
• e: base of the logarithm
• x: random variable
• λ :average rate

Poisson Probability Distribution

There are Some important List Of Top Mathematics Questions On Probability Asked In CBSE CLASS XII