Summary

1. Conditional Probability & Multiplication Theorem

• Conditional Probability: The probability of occurrence of event $A$ when it is known that event $B$ has already occurred is given by $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}, \qquad P(B)\ne 0$$ This formula reduces the sample space to event $B$ and then measures how much of that reduced space also belongs to $A$.

• Multiplication Rule: Rearranging the conditional probability formula gives $$P(A\cap B)=P(A)\,P(B\mid A)=P(B)\,P(A\mid B)$$ This result is extremely useful when events occur in stages.

• Independent Events: Two events $A$ and $B$ are said to be independent if the occurrence of one does not affect the probability of occurrence of the other. Mathematically, $$P(A\cap B)=P(A)P(B)$$ Equivalently, if $P(B)\ne 0$, then $$P(A\mid B)=P(A)$$ and if $P(A)\ne 0$, then $$P(B\mid A)=P(B)$$ Also, if $A$ and $B$ are independent, then the pairs $(A,B')$, $(A',B)$ and $(A',B')$ are also independent.

2. Bayes' Theorem & Total Probability

• Partition of Sample Space: A collection of events $E_1,E_2,\dots,E_n$ is said to form a partition of the sample space if $$E_i\cap E_j=\varnothing \quad (i\ne j)$$ and $$E_1\cup E_2\cup \cdots \cup E_n=S$$ with each $P(E_i)>0$. Thus the events are pairwise disjoint and exhaustive.

• Theorem of Total Probability: If $E_1,E_2,\dots,E_n$ form a partition of the sample space and $A$ is any event, then $$P(A)=\sum_{j=1}^{n} P(E_j)P(A\mid E_j)$$ This theorem is used when an event $A$ can happen through several mutually exclusive causes.

• Bayes' Theorem: If $E_1,E_2,\dots,E_n$ form a partition of the sample space and $A$ is an event with $P(A)>0$, then for any fixed $i$, $$P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_{j=1}^{n} P(E_j)P(A\mid E_j)}$$ Bayes' theorem helps us find the probability of a cause after observing the result. That is why it is often called the theorem of inverse probability.

3. Random Variables & Distributions

• Random Variable: A random variable is a real-valued function defined on the sample space. In school-level probability, we usually deal with discrete random variables.

• Mean (Expectation): If a random variable $X$ takes values $x_1,x_2,\dots,x_n$ with probabilities $p_1,p_2,\dots,p_n$, then its mean or expectation is $$E(X)=\mu=\sum x_i p_i$$ This gives the long-run average value of the random variable.

• Second Moment About Origin: $$E(X^2)=\sum x_i^2 p_i$$ This is used in the calculation of variance.

• Variance: Variance measures the spread or dispersion of the values of the random variable about its mean. It is given by $$Var(X)=\sigma^2=E(X^2)-[E(X)]^2$$ Since variance is a measure of spread, it is always non-negative.

• Standard Deviation: $$\sigma=\sqrt{Var(X)}$$ It is the positive square root of the variance and has the same unit as the random variable.

• Linear Properties: For constants $a$ and $b$, $$E(aX+b)=aE(X)+b$$ and $$Var(aX+b)=a^2Var(X)$$ Notice that adding a constant changes the mean but does not affect the variance, while multiplying by $a$ multiplies the variance by $a^2$.

4. Bernoulli Trials & Binomial Distribution

• Bernoulli Trials: A sequence of trials is called a sequence of Bernoulli trials if the following conditions are satisfied:

1. The number of trials is finite.
2. Each trial has exactly two possible outcomes, usually called success and failure.
3. The trials are independent.
4. The probability of success remains constant from trial to trial.

If the probability of success is $p$, then the probability of failure is $$q=1-p$$

• Binomial Distribution: If $X$ denotes the number of successes in $n$ Bernoulli trials, then $X$ follows a binomial distribution with parameters $n$ and $p$, written as $$X\sim B(n,p)$$ Its probability mass function is $$P(X=r)=\binom{n}{r}p^r q^{\,n-r}, \qquad r=0,1,2,\dots,n$$ where $$q=1-p$$

• Mean and Variance of Binomial Distribution: If $X\sim B(n,p)$, then $$E(X)=np$$ and $$Var(X)=npq$$ Therefore, the standard deviation is $$\sigma=\sqrt{npq}$$

Important Exam Tips for Board Exams

• Define Events Clearly: In questions based on Bayes' theorem or total probability, begin by writing statements such as: $$E_1=\text{first cause}, \qquad E_2=\text{second cause}, \qquad A=\text{observed event}$$ This makes the logic of your solution very clear.

• Use a Probability Distribution Table: For random variable questions, present the values in a neat tabular form with rows like $X$ and $P(X)$. This improves presentation and reduces mistakes.

• Show Combination Logic: In binomial and combinatorial probability problems, write at least one clear step showing how $$\binom{n}{r}$$ is obtained. This helps secure full method marks.

• Always Check Total Probability: For a valid probability distribution, $$\sum P(X=x_i)=1$$ If the sum is not 1, then some outcome has been omitted or a computational error has been made.

Important Exam Tips for JEE Main & Advanced

• Use the 'At Least One' Shortcut: Whenever you see the phrase at least one success, immediately think in terms of the complement: $$P(\text{at least one success})=1-P(\text{no success})$$ This is much faster than adding many separate cases.

• Remember Linearity of Expectation: For random variables $X$ and $Y$, $$E(X+Y)=E(X)+E(Y)$$ This is true even when $X$ and $Y$ are not independent. This fact is heavily used in advanced problems.

• Infinite-Turn Games: In turn-based games involving dice or coins, the total probability often forms an infinite geometric progression. Write the first winning term carefully and identify the common ratio correctly.

• Do Not Assume Independence Without Reason: If a question involves selection without replacement, then the outcomes are usually dependent. In such cases, use combinations or the multiplication rule instead of the independence formula.

• Check Binomial Conditions Before Applying Binomial Formula: The formula $$P(X=r)=\binom{n}{r}p^r q^{\,n-r}$$ can be used only when all Bernoulli trial conditions are satisfied.