Revision: Mathematics >> Probability CUET (UG) Probability

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

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

\[\mathrm{P(A/B)=\frac{P(A\cap B)}{P(B)}}\], where \[B\neq\phi\]

\[\mathrm{P(B/A)=\frac{P(A\cap B)}{P(A)}}\], where \[A\neq\phi\]

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.

Two events are said to be independent if the occurrence of one does not depend on the other.

If A and B are independent events, then

1. P(A/B) = P(A/B') = P(A)
2. P(B/A) = P(B/A') = P(B) 
3. If A and B are independent events, then 

a. P(A∩ B) = P(A). P (B) 

b. A and B' are also independent

c. A' and B' are also independent

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.

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 A and B are two events over the sample space S, then

1. P(A ∩ B) = P(B) · P (A/B)
2. P(A ∩ B) = P(A) · P (B/A)

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,

\[\mathrm{P(B_i/A)=\frac{P(B_i)P(A/B_i)}{\sum_{i=1}^nP(A\cap B_i)}}\]

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)