Revision: Statistics and Probability JEE Main Statistics and Probability

The arithmetic mean (or, simply, mean) of a set of numbers is obtained by dividing the sum of the numbers in the set by the number of numbers.

\[\mathbf{Mean}=\frac{\left(x_1+x_2+x_3+\ldots+x_n\right)}{n}=\frac{\Sigma x_i}{n}\] 

Median is the value of the middle-most observation(s). The median is a measure of central tendency which gives the value of the middle-most observation in the data.

The mode is the value of the observation that occurs most frequently; i.e., the observation with the maximum frequency is called the mode.

The probability distribution of the number of successes in an experiment consisting of n-Bernoulli trials obtained by the binomial expansion of (q + p )ⁿ is called the binomial distribution.

where p = probability of success and
q = probability of failure

\[P\left(X=r\right)=^{n}C_{r}p^{r}q^{n-r}\] is called probability function.

A set of favourable outcomes of a given sample space is an ‘event’. Event is a subset of the sample space.

• Events are denoted by capital letters A, B, C, D etc.
• The number of elements in event A is written n(A).
• Since A ⊆ S, always 0 ≤ n(A) ≤ n(S).
• Example: two coins tossed, A = getting at least one tail → A = {TT, TH, HT}, n(A) = 3.

The outcomes satisfying a particular condition are called favourable outcomes.

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:

If A and B are two events associated with the same sample space of a random experiment, the conditional probability of the event A given that B has occurred, i.e. P(A∣B), is given by

If a random variable X takes values x₁, x₂, …, xₙ with respective probabilities p₁, p₂, …, pₙ, then it is called the probability distribution of X.

A discrete random variable X is said to have the Poisson distribution with parameter m > 0, if its p.m. is given by

\[P(X=x)=\frac{e^{-m}m^x}{x!}\]  = 0, 1, 2, ....

A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies the following conditions:

• The trials are independent.
• The probability of success remains the same in all trials.

Direct Method:

\[\bar{x}=\frac{\sum f_ix_i}{\sum f_i}\]

where x_i​ = class mark, f_i​ = frequency

Short-cut (Assumed Mean) Method:

\[\bar{x} = A+\frac{\sum f_id_i}{\sum f_i}\]

where d_i = x_i - A
A is the assumed mean

Step-deviation Method:

\[\bar{x}=a+h\frac{\sum f_iu_i}{\sum f_i}\]

where \[u_i=\frac{x_i-a}{h}\]

h is the class width / common factor

If the number of data points (n) is odd, the median is,

Median = `((n+1)/2)^(th)` term

If n is even, the median is the average of the values at positions

Median = Average of  `(n/2)^(th)` and `(n/2+1)^(th)` values

G.M. between a and b

G² = ab 

G =\[\sqrt{ab}\]

G is the geometric mean between a and b.

\[P(A | B) = \frac{P(A \cap B)}{P(B)}\], 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: \[P(A' | B) = 1 - P(A | B)\], P(B) ≠ 0

For two events:

• \[P(E \cap F) = P(F) \cdot P(E | F)\],\[ P(F) \neq 0. \]

• \[P(E \cap F) = P(E) \cdot P(F | E)\],\[ P(E) \neq 0. \]

For three events:

If B₁, B₂,..., B_n 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,

• 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.

• 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).

• The sample space \[ S \] contains all possible outcomes.
• An event is a subset of the sample space.
• A simple event has one outcome; a compound event has more than one outcome.
• A certain event has probability \[ 1; \] an impossible event has probability \[ 0. \]
• The complement of \[ A \] is \[ A', \] and \[ P(A') = 1 - P(A). \]
• Mutually exclusive events cannot occur together: \[ A \cap B = \varnothing. \]
• Exhaustive events together cover the entire sample space: \[ A \cup B = S. \]
• For equally likely outcomes, \[ P(A) = \frac{n(A)}{n(S)}. \]
• For any two events, \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]

• 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.

• Conditional probability means probability under a given condition.

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

• First restrict the sample space according to the given condition, then count the favourable outcomes.

• 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), provided P(B) ≠ 0.