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.

Probability measures the degree of certainty of the occurrence of an event.

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:

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

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

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,

• 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).

Playing Cards – Key Facts

• Total cards = 52

• Red cards = 26 (Hearts, Diamonds)

• Black cards = 26 (Clubs, Spades)

• Each suit has 13 cards

• Face cards = King, Queen, Jack (Total = 12)

Properties:

• Complement Rule
  P(A′) = 1 − P(A)
  ⇒ P(A) + P(A′) = 1
• Range of Probability
  0 ≤ P(A) ≤ 1
• Impossible Event
  P(ϕ) = 0
• Certain Event
  P(S) = 1
• Subset Rule
  If A ⊆ B, then P(A) ≤ P(B)
• Difference of Events
  P(A ∩ B′) = P(A) − P(A ∩ B)
  P(A′ ∩ B) = P(B) − P(A ∩ B)
• Union of Two Events
  P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Union of Three Events
  P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
  − P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C)
• Mutually Exclusive Events (2 events)
  If A ∩ B = 0, then
  P(A ∪ B) = P(A) + P(B)
• Mutually Exclusive Events (multiple)
  P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ)
• Upper Bound of Union
  P(A ∪ B) ≤ P(A) + P(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 \[P(B) \neq 0\].

• 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).