Probability Notes for NIMCET, JEE, CUET MCA
NIMCET Notes and JEE Mains Notes are designed to provide clear, exam-focused theory, formulas, and shortcuts for all important chapters. These notes help students understand fundamental concepts, practice high-level problems, and revise efficiently before the exam. Whether you are preparing for MCA entrance exams or engineering entrance tests, these nimcet notes and jee mains notes cover definitions, formulas, solved examples, and chapter-wise summaries in a simple and structured manner so that students can prepare faster and score higher.1. Basic Terms
Sample Space (S): Set of all possible outcomes.
Event (E): Any subset of sample space.
Favourable Outcomes: Outcomes belonging to the event.
Probability of an event:
$$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}$$
Example
Find the probability of getting a head in a coin toss: $$P(H)=\frac{1}{2}$$
2. Types of Events
(a) Equally Likely Events
All outcomes have equal chance.
(b) Mutually Exclusive Events
If two events cannot occur at the same time:
$$P(A \cap B) = 0$$
(c) Exhaustive Events
Events covering whole sample space.
(d) Independent Events
Occurrence of one event does not affect the other. $$P(A \cap B)=P(A)\,P(B)$$
3. Addition Rule of Probability
For any events A and B:
$$P(A \cup B)=P(A)+P(B)-P(A\cap B)$$
Special Case (Mutually Exclusive)
If events do not occur together: $$P(A \cup B)=P(A)+P(B)$$
4. Multiplication Rule
For any events: $$P(A\cap B)=P(A)\cdot P(B|A)$$
Independent Case
If A and B are independent: $$P(A\cap B)=P(A)\,P(B)$$
5. Conditional Probability
Probability of A given that B has happened: $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$
Example
Two cards drawn from a deck. If first is Ace, find probability second is also Ace: $$P(A_2|A_1)=\frac{3}{51}$$
6. Bayes' Theorem
Used to reverse conditional probabilities. $$P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|\bar{A})P(\bar{A})}$$
7. Random Variables
A variable representing numerical value of an outcome. Types: • Discrete • Continuous
Probability Mass Function (PMF)
For discrete X: $$P(X=x) = p(x)$$ and $$\sum p(x)=1$$
Expectation (Mean)
$$E(X)=\sum x\,p(x)$$
Variance
$$Var(X)=E(X^2)-E(X)^2$$
8. Standard Distributions
(a) Bernoulli Distribution
PMF: $$P(X=1)=p, \; P(X=0)=1-p$$ Mean: $p$ Variance: $p(1-p)$
(b) Binomial Distribution
$$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$$ Mean: $np$ Variance: $np(1-p)$
(c) Poisson Distribution
$$P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}$$ Mean = Variance = $\lambda$
(d) Uniform Distribution
$$f(x)=\frac{1}{b-a},\; a\le x\le b$$ Mean: $\frac{a+b}{2}$ Variance: $\frac{(b-a)^2}{12}$
9. Important Results
1. $0 \le P(E) \le 1$
2. $P(S)=1$, $P(\phi)=0$
3. $P(\bar{E})=1-P(E)$
4. $P(A\cap B)=P(A)+P(B)-P(A\cup B)$
5. For independent events: $P(A|B)=P(A)$
10. Practice Questions (Exam Level)
Q1. A die is rolled. Find probability that outcome is a multiple of 3.
Solution: Fav = {3,6}, Total = 6 → $$P=\frac{2}{6}=\frac{1}{3}$$
Q2. Two coins tossed. Probability of at least one head?
Total = 4, favourable = 3 → $$P=\frac{3}{4}$$
Q3. A bag has 3 red and 5 blue balls. 2 drawn. Probability both red?
$$P=\frac{\binom{3}{2}}{\binom{8}{2}}=\frac{3}{28}$$
Q4. If $P(A)=0.4$, $P(B)=0.5$, $P(A\cap B)=0.2$ find $P(A\cup B)$.
$$P=0.4+0.5-0.2=0.7$$
Q5. In a binomial distribution $n=5$, $p=0.4$, find $P(X=2)$.
$$P=\binom52(0.4)^2(0.6)^3=0.3456$$
Probability Inequalities (NIMCET Notes and JEE Mains Notes)
These probability inequalities are useful for competitive exams like NIMCET and JEE Mains. You can directly use this content in your nimcet notes and jee mains notes.
1. Basic Probability Range
For any event $A$: $$0 \le P(A) \le 1$$
2. Complement Inequality
For an event $A$ and its complement $\bar{A}$: $$P(A) + P(\bar{A}) = 1$$ Hence, $$P(\bar{A}) = 1 - P(A)$$ and $$0 \le P(\bar{A}) \le 1$$
3. Union and Intersection Inequalities
For any two events $A$ and $B$:
Union always increases possibilities: $$P(A \cup B) \ge P(A), \quad P(A \cup B) \ge P(B)$$
Intersection always reduces possibilities: $$P(A \cap B) \le P(A), \quad P(A \cap B) \le P(B)$$
4. Addition Inequality for Two Events
Using the inclusion–exclusion principle: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Since $P(A \cap B) \ge 0$: $$P(A \cup B) \le P(A) + P(B)$$
Also, as $P(A \cup B) \ge 0$ and $P(A \cap B) \le 1$: $$P(A) + P(B) - 1 \le P(A \cup B) \le P(A) + P(B)$$
5. Inequality for Union of n Events
For events $A_1, A_2, \dots, A_n$: $$P(A_1 \cup A_2 \cup \dots \cup A_n) \le \sum_{i=1}^{n} P(A_i)$$ This is a general form of the union bound.
6. Boole’s Inequality (Union Bound)
For any number of events $A_1, A_2, A_3, \dots$: $$P\left(\bigcup_{i=1}^{\infty} A_i\right) \le \sum_{i=1}^{\infty} P(A_i)$$ This is very useful when the intersections are not known.
7. Bonferroni Inequality (Two Events)
For two events $A$ and $B$: $$P(A) + P(B) - 1 \le P(A \cap B)$$ because $P(A \cup B) \le 1$ and $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
8. Markov’s Inequality
If $X$ is a non-negative random variable and $a > 0$, then: $$P(X \ge a) \le \frac{E(X)}{a}$$ This gives an upper bound on the tail probability using only the expectation.
9. Chebyshev’s Inequality
If $X$ is a random variable with mean $\mu$ and variance $\sigma^2$, then for any $k > 0$: $$P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}$$ This inequality shows that most of the probability mass lies within a few standard deviations of the mean.
10. Jensen’s Inequality (Advanced)
Let $X$ be a random variable and $\phi$ be a convex function. Then: $$\phi(E(X)) \le E(\phi(X))$$ This inequality is widely used in advanced probability and statistics, information theory, and optimization.
11. Quick Summary
- $0 \le P(A) \le 1$
- $P(\bar{A}) = 1 - P(A)$
- $P(A \cup B) \le P(A) + P(B)$
- $P(A_1 \cup \dots \cup A_n) \le \sum P(A_i)$
- $P(A) + P(B) - 1 \le P(A \cap B)$
- Markov: $P(X \ge a) \le \dfrac{E(X)}{a}$
- Chebyshev: $P(|X - \mu| \ge k\sigma) \le \dfrac{1}{k^2}$
- Jensen (convex $\phi$): $\phi(E(X)) \le E(\phi(X))$