Probability — JEE Mains Maths Formulas and Traps

Counting Favourable Outcomes

Selections: counting with combinations

Classical probability

P(E)=\frac{n(E)}{n(S)}

Ordered outcomes: words, sequences and orders

Order by symmetry

P(\text{given relative order of }k\text{ items})=\frac{1}{k!}

Common traps

Same sample space in both counts

If the favourable outcomes are counted as ordered pairs, the sample space must be ordered pairs too. Mixing

\binom n2

with

n(n-1)

doubles or halves the answer.

Include non-integer ratios

A G.P. of whole numbers can have ratio

\frac32

or

\frac43

:

(4,6,9)

,

(9,12,16)

. Counting only whole-number ratios undercounts.

Dice, Digits and Divisibility

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Dice and weighted outcomes

Independent throws

P(\text{result})=\sum_{\text{favourable }(a,b)}P(a)\,P(b)

Numbers with divisibility or digit properties

Euler's function

\varphi(N)=N\prod_{p\mid N}\left(1-\frac1p\right)

Common traps

Ordered pairs

(1,3)

and

(3,1)

are different outcomes on two dice. Listing unordered pairs undercounts every mixed pair by half.

The leading digit cannot be 0

In a random

k

-digit number the first digit is 1–9, so it is odd with probability

\frac59

, not

\frac12

. The other digits are 0–9.

Random Coefficients and Inequalities

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Quadratics with random coefficients

Discriminant conditions

ax^2+bx+c:\quad b^2-4ac\ \gtreqless\ 0

Other conditions on random outcomes

Sum of two dice

P(N=k)=\frac{6-|k-7|}{36},\quad k=2,\dots,12

Common traps

Strict or not

'Real roots' allows

b^2=4ac

; 'two distinct real roots' and 'one root bigger than the other' do not. Check whether the equality cases belong in the count.

Integer endpoints

After solving the inequality, check whether the endpoints are integers and whether they are included; a strict inequality drops them.

Addition, Conditional Probability and Independence

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Addition rule and conditional probability

Conditional probability

P(A\mid B)=\frac{P(A\cap B)}{P(B)}

Independent events

Independence

P(A\cap B)=P(A)\,P(B)

Trials repeated until a success

Alternate turns

P(A\text{ wins})=p_A+q_Aq_Bp_A+(q_Aq_B)^2p_A+\cdots=\frac{p_A}{1-q_Aq_B}

Common traps

Condition on the complement

P(A\mid B')=\frac{P(A)-P(A\cap B)}{1-P(B)}

. Dividing by

P(B)

out of habit, or forgetting to remove

A\cap B

, are the two usual slips.

Take the root that is a probability

Equations from 'exactly one' are quadratic in

p

. Both roots may lie in

(0,1)

, but each event's probability must too: with

P(B)=2p

, need

p\le\frac12

.

Who throws first

The player who throws first has the extra head start: their series starts at

p_A

, the other's at

q_Ap_B

. Swapping them gives the other player's chance.

Total Probability and Bayes' Theorem

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Total probability

P(E)=\sum_{i}P(H_i)\,P(E\mid H_i)

Bayes' theorem

P(H_k\mid E)=\frac{P(H_k)\,P(E\mid H_k)}{\sum_iP(H_i)\,P(E\mid H_i)}

Inferring an unknown bag or a lost card

Weights for an unknown bag

P(k\mid\text{draw})=\frac{\binom kr}{\sum_j\binom jr}\quad(\text{equal priors})

Common traps

The receiving bag has one more ball

After a transfer, the second bag holds one extra ball. Using its original total in the denominator is the usual mistake.

Priors are not always equal

When bags or machines are chosen with different chances, keep

P(H_i)

in every product. Dropping them is only allowed when they are equal.

Include the impossible compositions

Compositions that could not produce the draw have weight 0 but still count when setting equal priors. The denominator sums over all of them; the zeros simply add nothing.

Binomial Distribution

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Binomial probabilities

Binomial probability

P(X=r)=\binom nr\,p^{r}q^{\,n-r}

Recovering n and p from the mean and variance

Moments

E(X)=np,\qquad\operatorname{Var}(X)=npq

Common traps

Identify the trial and its p

When a pair of dice is thrown, one throw of the pair is one trial; its success probability comes from counting outcomes of the pair (sum 5 has probability

\frac19

), not from one die.

The mean is the larger root

Since

q<1

, the variance is less than the mean. When the two come from a quadratic, assigning the smaller root to the mean gives

q>1

, which is impossible.

Random Variables: Mean and Variance

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Distributions, mean and variance

Variance

\operatorname{Var}(X)=E(X^2)-\big(E(X)\big)^2

Special items in a sample without replacement

Sampling without replacement

E(X)=\frac{nK}{N},\qquad\operatorname{Var}(X)=\frac{nK}{N}\cdot\frac{N-K}{N}\cdot\frac{N-n}{N-1}

Common traps

Square the values, not the probabilities

E(X^2)=\sum x^2P(x)

. Squaring

P(x)

, or using

(\sum xP)^2

in place of

E(X^2)

, gives a wrong variance.

Without replacement changes the variance

The mean is the same with or without replacement, but the variance is smaller without: it carries the factor

\frac{N-n}{N-1}

.

Probability formulas

Counting Favourable Outcomes

Selections: counting with combinations

Ordered outcomes: words, sequences and orders

Common traps

Dice, Digits and Divisibility

Dice and weighted outcomes

Numbers with divisibility or digit properties

Common traps

Random Coefficients and Inequalities

Quadratics with random coefficients

Other conditions on random outcomes

Common traps

Addition, Conditional Probability and Independence

Addition rule and conditional probability

Independent events

Trials repeated until a success

Common traps

Total Probability and Bayes' Theorem

Inferring an unknown bag or a lost card

Common traps

Binomial Distribution

Binomial probabilities

Recovering n and p from the mean and variance

Common traps

Random Variables: Mean and Variance

Distributions, mean and variance

Special items in a sample without replacement

Common traps

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