IIT JEE Test Series — Practice smarter, perform stronger.

Probability formulas

Master Probability through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

Practise these formulas Know more about IIT JEE

Back to Formula Book

Probability, every formula

34 formulas, typeset and free. Print it, or keep it open beside your practice.

Classical definition
$P (A) = \frac{n ( A )}{n ( S )} = \frac{m}{n}$
favourable / total
Q1MCQClassical
A die is rolled. The probability of getting an even number is:
- A
  $\dfrac12$
- B
  $\dfrac13$
- C
  $\dfrac16$
- D
  $\dfrac23$
Range of probability
$0 \leq P (A) \leq 1$
Q1MCQRange
Which of these cannot be a probability?
- A
  $1.2$
- B
  $0$
- C
  $0.5$
- D
  $1$
Complement
$P (\overset{ˉ}{A}) = 1 - P (A)$
Impossible & certain events
$P (\emptyset) = 0, P (S) = 1$
Odds in favour of A
$P (A) : P (\overset{ˉ}{A})$
Q1MCQOdds in favour
The odds in favour of $A$ are defined as:
- A
  $P(A):P(\bar A)$
- B
  $P(\bar A):P(A)$
- C
  $P(A):1$
- D
  $P(A)\cdot P(\bar A)$
Probability from odds x:y
$P (A) = \frac{x}{x + y}, P (\overset{ˉ}{A}) = \frac{y}{x + y}$
Q1MCQProbability from odds
If the odds in favour of $A$ are $2:3$, then $P(A)$ is:
- A
  $\dfrac25$
- B
  $\dfrac35$
- C
  $\dfrac23$
- D
  $\dfrac12$
Addition theorem
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Q1MCQAddition theorem
If $P(A)=0.5,\ P(B)=0.4,\ P(A\cap B)=0.2$, then $P(A\cup B)$ is:
- A
  $0.7$
- B
  $0.9$
- C
  $1.1$
- D
  $0.6$
Mutually exclusive addition
$P (A \cup B) = P (A) + P (B)$
A∩B=∅
Addition for three events
$P (A \cup B \cup C) = \sum P (A) - \sum P (A \cap B) + P (A \cap B \cap C)$
Q1MCQThree events
$P(A\cup B\cup C)$ includes the term:
- A
  $+P(A\cap B\cap C)$
- B
  $-P(A\cap B\cap C)$
- C
  $+2P(A\cap B\cap C)$
- D
  no triple term
Exactly one of A, B
$P (A) + P (B) - 2 P (A \cap B)$
Q1MCQExactly one
$P(\text{exactly one of }A,B)$ equals:
- A
  $P(A)+P(B)-2P(A\cap B)$
- B
  $P(A)+P(B)-P(A\cap B)$
- C
  $P(A)+P(B)$
- D
  $P(A\cap B)$
De Morgan (neither)
$P (\overset{ˉ}{A} \cap \overset{ˉ}{B}) = 1 - P (A \cup B)$
De Morgan (not both)
$P (\overset{ˉ}{A} \cup \overset{ˉ}{B}) = 1 - P (A \cap B)$
Conditional probability
$P (B ∣ A) = \frac{P ( A \cap B )}{P ( A )}, P (A) \neq = 0$
Multiplication theorem
$P (A \cap B) = P (A) P (B ∣ A) = P (B) P (A ∣ B)$
Q1MCQMultiplication
$P(A\cap B)$ equals:
- A
  $P(A)\,P(B\mid A)$
- B
  $P(A)+P(B)$
- C
  $P(A)\,P(B)$ always
- D
  $P(A\mid B)$
Independent events
$P (A \cap B) = P (A) P (B)$
Independent — union
$P (A \cup B) = 1 - P (\overset{ˉ}{A}) P (\overset{ˉ}{B})$
Independence of complements
$A, B independent \Rightarrow \overset{ˉ}{A}, \overset{ˉ}{B} independent$
Q1MCQComplement independence
If $A,B$ are independent, then $\bar A,\bar B$ are:
- A
  independent
- B
  mutually exclusive
- C
  equal
- D
  dependent
Law of total probability
$P (A) = i \sum P (E_{i}) P (A ∣ E_{i})$
E_i partition of S
Q1MCQTotal probability
The law of total probability expresses $P(A)$ as:
- A
  $\sum_i P(E_i)P(A\mid E_i)$
- B
  $\prod_i P(E_i)$
- C
  $\sum_i P(A\cap E_i)^2$
- D
  $P(A\mid E_1)$
Bayes' theorem
$P (E_{i} ∣ A) = \frac{P ( E _{i} ) P ( A ∣ E _{i} )}{\sum _{j} P ( E _{j} ) P ( A ∣ E _{j} )}$
Q1MCQBayes' theorem
Bayes' theorem gives $P(E_i\mid A)$ as:
- A
  $\dfrac{P(E_i)P(A\mid E_i)}{\sum_j P(E_j)P(A\mid E_j)}$
- B
  $P(E_i)P(A\mid E_i)$
- C
  $\dfrac{P(A\mid E_i)}{P(E_i)}$
- D
  $P(A\cap E_i)$
Geometric probability
$P = \frac{measure of favourable region}{measure of whole region}$
Q1MCQGeometric
Geometric probability is the ratio of:
- A
  measure of favourable region to whole region
- B
  favourable outcomes to total outcomes
- C
  area to perimeter
- D
  length to volume
n coins: exactly r heads
$P = \frac{^{n} C _{r}}{2 ^{n}}$
Q1MCQCoins exactly r heads
Three fair coins are tossed. The probability of exactly $2$ heads is:
- A
  $\dfrac38$
- B
  $\dfrac18$
- C
  $\dfrac12$
- D
  $\dfrac14$
n coins: at least one head
$P = 1 - \frac{1}{2 ^{n}}$
Q1MCQAt least one head
Two fair coins are tossed. The probability of at least one head is:
- A
  $\dfrac34$
- B
  $\dfrac14$
- C
  $\dfrac12$
- D
  $1$
Two dice: sum k (favourable)
$k - 1 (2 \leq k \leq 7), 13 - k (8 \leq k \leq 12)$
out of 36
Q1MCQTwo dice sum
When two dice are rolled, the probability of getting a sum of $7$ is:
- A
  $\dfrac16$
- B
  $\dfrac{5}{36}$
- C
  $\dfrac{1}{12}$
- D
  $\dfrac{7}{36}$
Derangement probability (n envelopes)
$P = \frac{1}{n !} (all correct), 1 - \frac{1}{n !} (at least one wrong)$
Q1MCQEnvelopes
The probability that all $3$ letters go into their correct envelopes is:
- A
  $\dfrac16$
- B
  $\dfrac13$
- C
  $\dfrac12$
- D
  $1$
Binomial probability
$P (X = r) =^{n} C_{r} p^{r} q^{n - r}, q = 1 - p$
Q1MCQBinomial probability
For $n=4,\ p=\dfrac12$, the probability of exactly $2$ successes is:
- A
  $\dfrac{3}{8}$
- B
  $\dfrac14$
- C
  $\dfrac12$
- D
  $\dfrac{1}{16}$
Binomial mean
$μ = n p$
Binomial variance
$σ^{2} = n pq$
Binomial S.D.
$σ = n pq$
Binomial: mean > variance
$mean > variance (∵ q < 1)$
Q1MCQMean vs variance
In a binomial distribution, the mean is always:
- A
  greater than the variance
- B
  less than the variance
- C
  equal to the variance
- D
  zero
Most probable number of successes
$r = ⌊(n + 1) p ⌋ (mode)$
Q1MCQMost probable successes
For $n=5,\ p=\dfrac12$, the most probable number of successes is:
- A
  $3$
- B
  $2$
- C
  $5$
- D
  $1$
Mean of a random variable
$μ = E (X) = \sum x_{i} P (x_{i})$
Variance of a random variable
$σ^{2} = \sum x_{i}^{2} P (x_{i}) - μ^{2}$
Q1MCQVariance of r.v.
The variance of a random variable equals:
- A
  $\sum x_i^{2}P(x_i)-\mu^{2}$
- B
  $\sum x_iP(x_i)$
- C
  $\mu^{2}$
- D
  $\sum x_i^{2}P(x_i)$
Sum of probabilities
$i \sum P (x_{i}) = 1$
P(A∩B) when A⊆B
$A \subseteq B \Rightarrow P (A) \leq P (B)$
Q1MCQSubset events
If $A\subseteq B$, then:
- A
  $P(A)\le P(B)$
- B
  $P(A)\ge P(B)$
- C
  $P(A)=P(B)$
- D
  $P(A)+P(B)=1$

More JEE Advanced Mathematics formula sheets

Every chapter sheet is typeset, free and open without a sign-in.

Other ways to revise this chapter

Master this chapter with similar other learning materials.

Preparing students for India’s top institutes

Our students are currently into top technological and medical institutes of India.

IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela
IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela

Join QuestPix, Today!

Get notified first, with exam & curriculum updates, course & test series launch offers, motivation & success stories and free learning resources recommended by toppers.

Probability formulas

Probability, every formula

Classical definition

Range of probability

Complement

Impossible & certain events

Odds in favour of A

Probability from odds x:y

Addition theorem

Mutually exclusive addition

Addition for three events

Exactly one of A, B

De Morgan (neither)

De Morgan (not both)

Conditional probability

Multiplication theorem

Independent events

Independent — union

Independence of complements

Law of total probability

Bayes' theorem

Geometric probability

n coins: exactly r heads

n coins: at least one head

Two dice: sum k (favourable)

Derangement probability (n envelopes)

Binomial probability

Binomial mean

Binomial variance

Binomial S.D.

Binomial: mean > variance

Most probable number of successes

Mean of a random variable

Variance of a random variable

Sum of probabilities

P(A∩B) when A⊆B

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

Join QuestPix, Today!