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.

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

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.

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, ....

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.

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)