JEE WhatsApp Group
Probability and Statistics JEE Notes
Probability and Statistics together carry 7-10 marks in JEE Main and a steady two problems in JEE Advanced. Concepts range from basic classical probability to random variables, and from mean/median to standard deviation. Questions are usually moderate if you know the definitions and can set up events or frequency tables quickly. These comprehensive JEE Maths Notes walk you through each idea with neat derivations and exam-level illustrations.
Probability Theory and Random Variables
Essential Concepts
- Random experiment, sample space $$S$$, event $$E\subseteq S$$.
- Classical (equally likely) probability: $$P(E)=\dfrac{|E|}{|S|}$$.
- Complementary, mutually exclusive and exhaustive events.
- Conditional probability: $$P(A|B)=\dfrac{P(A\cap B)}{P(B)}$$ when $$P(B)\neq0$$.
- Independence: $$P(A\cap B)=P(A)P(B)$$.
- Total probability and Bayes’ theorem.
Discrete Random Variables
A variable $$X$$ taking finite or countably infinite values $$x_i$$ with probabilities $$p_i$$ where $$\sum p_i=1$$.
- Expectation: $$E[X]=\sum x_i p_i$$
- Variance: $$Var(X)=E[X^2]-\{E[X]\}^2$$
Important Distributions for JEE
| Distribution | Parameters | PMF / PDF | Mean | Variance |
|---|---|---|---|---|
| Binomial | $$n,p$$ | $$P(X=r)=\binom{n}{r}p^{\,r}(1-p)^{\,n-r}$$ | $$np$$ | $$np(1-p)$$ |
| Poisson | $$\lambda$$ | $$P(X=r)=\dfrac{\lambda^{\,r}e^{-\lambda}}{r!}$$ | $$\lambda$$ | $$\lambda$$ |
| Uniform (discrete) | $$k$$ outcomes | $$1/k$$ each | $$\dfrac{k+1}{2}$$ | $$\dfrac{k^2-1}{12}$$ |
Worked Example
Question: A fair die is thrown four times. Find the probability of getting exactly two sixes.
Solution: Here $$n=4, p=1/6, r=2$$.
$$P=\binom{4}{2}\left(\frac16\right)^2\left(\frac56\right)^2=\frac{6\times25}{1296}=\frac{150}{1296}=\frac{25}{216}.$$
Continuous Random Variables (JEE Advanced focus)
- Probability density function $$f(x)$$ with $$\int_{-\infty}^{\infty}f(x)\,dx=1$$.
- Cumulative distribution function $$F(x)=\int_{-\infty}^{x}f(t)\,dt$$.
- Uniform $$U(a,b)$$ and Normal $$N(\mu,\sigma^2)$$ basics. Use standard normal tables when required.
Statistics: Descriptive Measures and Data Analysis
Measures of Central Tendency
- Arithmetic mean $$\bar{x}=\dfrac{\sum x_i}{n}$$.
- Median for odd $$n$$: middle value after sorting; for even $$n$$: mean of two central values.
- Mode: value with maximum frequency; for grouped data, use $$l + \dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$$.
Measures of Dispersion
- Range $$=x_{\max}-x_{\min}$$.
- Variance (ungrouped) $$=\dfrac{\sum(x_i-\bar{x})^2}{n}$$, standard deviation $$\sigma=\sqrt{Var}$$.
- Coefficient of variation $$CV=\dfrac{\sigma}{\bar{x}}\times100\%$$ – handy to compare consistency.
Correlation and Regression (occasionally asked)
- Pearson’s correlation coefficient $$r=\dfrac{\sum (x-\bar{x})(y-\bar{y})}{n\sigma_x\sigma_y}$$, $$-1\le r\le1$$.
- Regression line of $$y$$ on $$x$$: $$y-\bar{y}=r\dfrac{\sigma_y}{\sigma_x}(x-\bar{x})$$.
Example on Standard Deviation
Question: The marks of five students are 10, 12, 15, 18, 20. Calculate $$\sigma$$.
- Mean $$\bar{x}=75/5=15$$.
- Sum of squared deviations $$=(10-15)^2+(12-15)^2+(15-15)^2+(18-15)^2+(20-15)^2=25+9+0+9+25=68$$.
- Variance $$=68/5=13.6$$; hence $$\sigma=\sqrt{13.6}\approx3.69$$.
Important Formulas and Results at a Glance
| Concept | Formula | Quick Tip |
|---|---|---|
| Addition rule | $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ | For mutually exclusive events, just add. |
| Conditional probability | $$P(A|B)=\dfrac{P(A\cap B)}{P(B)}$$ | Always check $$P(B)\neq0$$. |
| Total probability | $$P(B)=\sum_i P(A_i)P(B|A_i)$$ | $$A_i$$’s must be mutually exclusive and exhaustive. |
| Bayes’ theorem | $$P(A_i|B)=\dfrac{P(A_i)P(B|A_i)}{\sum_j P(A_j)P(B|A_j)}$$ | Useful in repeated coin/dice questions. |
| Mean of grouped data | $$\bar{x}=\dfrac{\sum f_ix_i}{\sum f_i}$$ | Take assumed mean to speed up. |
| Standard deviation (grouped) | $$\sigma=\sqrt{\dfrac{\sum f_i(x_i-\bar{x})^2}{\sum f_i}}$$ | Combine classes if frequencies tiny. |
| Coefficient of variation | $$CV=\dfrac{\sigma}{\bar{x}}\times100$$ | Lower CV ⇒ more consistent. |
JEE Important Points, Common Mistakes and Quick Revision
- Write events clearly before plugging into formulas; half of the errors happen here.
- For “at least” questions on binomial, use complement $$1-P(0)-P(1)-\dots$$ to save time.
- Remember that independence implies in both directions; mere disjointness does not.
- Never round off mean midway while computing $$\sigma$$ – keep one extra decimal.
- In Bayes’ problems, list the complete event tree; missing one branch gives wrong denominator.
- For grouped data, use class mark $$x_i=\dfrac{lower+upper}{2}$$ only.
- Practice past JEE Questions on random variables — patterns repeat.
- Take at least two timed JEE Mains Mock Test papers for speed with cumulative distribution tables.
Probability and Statistics JEE Notes: Conclusion
Probability and Statistics reward clarity: define events, pick the right distribution, and apply the standard formulas without missing conditions. Keep the summary sheet handy, revise the worked examples, and drill past papers. With systematic practice via JEE Mains Online Coaching and swift reference to JEE Formula Sheets, these 7-10 marks are yours for the taking.
.webp)
