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

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.

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 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.

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

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

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,

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)

• 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).

• 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).

• 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.