Unit 4 · 10 Marks
Topics & Key Formulas
Four topics examined across MCQs, short answers, long answers, and case studies. All aligned to CBSE 2026–27.
Class 12 Applied Maths Unit 4 — Probability Distributions Study Material
This page covers all topics in Unit 4 of CBSE Class 12 Applied Mathematics — a 10-mark unit. You'll find 33 interactive MCQs, 12 short answer and 5 long answer solved examples, and 3 case studies on Random Variables, Binomial, Poisson, and Normal Distributions. Aligned to CBSE Applied Maths Syllabus 2026-27.
Random VariableBinomial DistributionPoisson DistributionNormal DistributionCBSE 2026–27
1. Random Variable & Probability Distribution Table
Discrete and continuous random variables, expectation E(X), variance, constructing probability distribution tables. Always verify \(\sum P(X=x) = 1\).
\(E(X) = \sum x \cdot P(X=x)\)
\(\text{Var}(X) = E(X^2) - [E(X)]^2\)
2. Binomial Distribution
Bernoulli trials, binomial PMF, mean np, variance npq. Conditions: fixed n, independent trials, constant p.
\(P(X=x) = \binom{n}{x} p^x q^{n-x}\)
Mean \(= np\), Variance \(= npq\)
3. Poisson Distribution
Limiting form of binomial (large n, small p). Mean = Variance = λ. Applications: calls per minute, defects per batch.
\(P(X=x) = \dfrac{e^{-\lambda}\lambda^x}{x!}\)
Mean \(= \lambda\), SD \(= \sqrt{\lambda}\)
4. Normal Distribution
Bell curve, symmetric about mean. Convert to standard normal using z-score. Use given area tables in exam.
\(Z = \dfrac{X - \mu}{\sigma}\)
\(\text{Var}(aX+b) = a^2\,\text{Var}(X)\) (b has no effect)
4
Probability Distributions10 Marks · Series of 3 Videos
Probability Distributions — Complete Series (3 Videos)
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- Unit 1 — Numbers & Quantification
- Unit 2 — Algebra (Matrices)
- Unit 3 — Calculus
- Unit 4 — Probability Distributions ✓
- Unit 5 — Inferential Statistics
- Unit 6 — Index Numbers & Time Data
- Unit 7 — Financial Mathematics
- Unit 8 — Linear Programming
Interactive Practice
Practice MCQs — Unit 4
Select your answer, then click Show Answer to check and reveal the full explanation.
Question 1CUET 2022
For a random variable \(X\), \(E(X)=3\) and \(E(X^2)=11\). What is the variance of \(X\)?
Formula: \(\text{Var}(X)=E(X^2)-[E(X)]^2=11-9=\mathbf{2}\)
Question 2CUET 2022
For events \(A\) and \(B\): \(P(A)=\tfrac{1}{4}\), \(P(B|A)=\tfrac{1}{2}\), \(P(A|B)=\tfrac{1}{3}\). What is \(P(B)\)?
Step 1: \(P(A\cap B)=P(B|A)\times P(A)=\tfrac{1}{2}\times\tfrac{1}{4}=\tfrac{1}{8}\)
Step 2: \(P(B)=\dfrac{P(A\cap B)}{P(A|B)}=\dfrac{1/8}{1/3}=\dfrac{3}{8}\)
Step 2: \(P(B)=\dfrac{P(A\cap B)}{P(A|B)}=\dfrac{1/8}{1/3}=\dfrac{3}{8}\)
Question 3CUET 2022
A random variable \(X\) has the probability distribution \(P(X=0)=\tfrac{1}{5},\;P(X=1)=\tfrac{2}{5},\;P(X=2)=\tfrac{2}{5}\). What is \(E(X^2)\)?
\(E(X^2)=0\cdot\tfrac{1}{5}+1\cdot\tfrac{2}{5}+4\cdot\tfrac{2}{5}=0+\tfrac{2}{5}+\tfrac{8}{5}=\mathbf{2}\)
Question 4CUET 2022
A random variable \(X\) has the probability distribution \(P(X=0)=0.3,\;P(X=1)=0.4,\;P(X=2)=0.3\). What is \(E(X^2)\)?
\(E(X^2)=0(0.3)+1(0.4)+4(0.3)=0+0.4+1.2=\mathbf{1.6}\)
Question 5CUET 2022
A child wins ₹5 if all heads or all tails appear on a toss of 3 coins, and loses ₹3 otherwise. What is the expected amount the child loses per game?
\(P(\text{win})=\tfrac{2}{8}=\tfrac{1}{4}\). \(E=\tfrac{1}{4}(5)+\tfrac{3}{4}(-3)=\tfrac{5}{4}-\tfrac{9}{4}=-1\). Expected loss of ₹1 per game.
Question 6CUET 2022
A random variable \(X\) has probability function \(P(X=x)=\binom{4}{x}\!\left(\tfrac{1}{2}\right)^4\) for \(x=0,1,2,3,4\). What is the variance of \(X\)?
Binomial: \(n=4,\;p=q=\tfrac{1}{2}\). \(\text{Var}=npq=4\times\tfrac{1}{2}\times\tfrac{1}{2}=\mathbf{1}\)
Question 7CUET 2023
A box contains 100 bulbs, of which 10 are defective. If a sample of 5 bulbs is drawn, what is the probability that none of them is defective?
\(P(\text{good})=\tfrac{90}{100}=\tfrac{9}{10}\). For 5 independent selections: \(\left(\dfrac{9}{10}\right)^5\)
Question 8CUET 2023
What is the mean number of heads when a fair coin is tossed twice?
\(X\sim B(2,\tfrac{1}{2})\). Mean \(=np=2\times\tfrac{1}{2}=\mathbf{1}\)
Question 9CUET 2023
In a survey, 70% of members favour a proposal. Let \(X=1\) if a member is in favour and \(X=0\) if opposed. What is \(E(X^2)\)?
\(E(X^2)=1^2\times0.7+0^2\times0.3=\mathbf{0.7}\). For Bernoulli \(X\in\{0,1\}\): \(E(X^2)=E(X)=p\).
Question 10CUET 2023
The average number of calls received per minute between 3 pm and 5 pm follows a Poisson distribution with mean 5. What is the probability of receiving exactly one call in a given minute?
\(\lambda=5\). \(P(X=1)=\dfrac{e^{-5}\cdot5^1}{1!}=\mathbf{5e^{-5}}\)
Question 11CUET 2023
For a binomial distribution, the sum and product of the mean and variance are 18 and 72 respectively. What is \(P(X\le1)\)?
Official answer: \(\dfrac{37}{729}\) — accept as per CBSE key. \(np+npq=18\) and \(np\cdot npq=72\) gives \(np=12,npq=6,q=\tfrac{1}{2},p=\tfrac{1}{2},n=24\).
Question 12CUET 2023
A random variable \(X\) has probability function \(P(X=x)=k(x+1)\) for \(x=1,2,3,4,5\). What is the value of \(k\)?
Sum=1: \(k(2+3+4+5+6)=21k=1\Rightarrow k=\dfrac{1}{21}\)
Question 13CUET 2023
Match the properties of Poisson Distribution in List I with their correct values in List II:
Choose the correct matching from the options below:
Correct match: A–III, B–I, C–IV, D–II
A. Variance of Poisson \(= \lambda\) → III
B. SD of Poisson \(= \sqrt{\lambda}\) → I
C. SD when mean = 4: \(\sqrt{4} = 2\) → IV
D. Variance when mean = 4: \(\lambda = 4\) → II
Key fact: For Poisson distribution, Mean = Variance = \(\lambda\), so SD \(= \sqrt{\lambda}\).
A. Variance of Poisson \(= \lambda\) → III
B. SD of Poisson \(= \sqrt{\lambda}\) → I
C. SD when mean = 4: \(\sqrt{4} = 2\) → IV
D. Variance when mean = 4: \(\lambda = 4\) → II
Key fact: For Poisson distribution, Mean = Variance = \(\lambda\), so SD \(= \sqrt{\lambda}\).
Question 14CBSE 2022
A biased die has the number 1 on three faces, the number 2 on two faces, and the number 5 on one face. What is \(E(X)\)?
\(E(X)=1\cdot\tfrac{3}{6}+2\cdot\tfrac{2}{6}+5\cdot\tfrac{1}{6}=\tfrac{3+4+5}{6}=\dfrac{12}{6}=\mathbf{2}\)
Question 15CBSE 2022
A coin is tossed 6 times and \(X=|\text{heads}-\text{tails}|\). What are the possible values of \(X\)?
\(X=|2H-6|\). For \(H=0..6\): \(X=6,4,2,0,2,4,6\). Unique values: {0,2,4,6} — always even.
Question 16CBSE 2022
A pair of dice is thrown 4 times. What is the mean of the distribution of the number of doublets obtained?
\(p=\tfrac{6}{36}=\tfrac{1}{6}\). Mean\(=np=4\times\tfrac{1}{6}=\dfrac{2}{3}\)
Question 17CBSE 2022
The mean of a binomial distribution is 81. In which interval does the standard deviation lie?
\(np=81\). Since \(0<q<1\): \(\sigma=\sqrt{npq}<\sqrt{81}=9\) and \(\sigma\ge0\). SD \(\in\mathbf{[0,9)}\).
Question 18
A random variable \(X\) has probability function \(P(X=x)=kx\) for \(x=1,2,3,4\). What is the value of \(k\)?
Sum=1: \(k(1+2+3+4)=10k=1\Rightarrow k=\dfrac{1}{10}\)
Question 19
What is the mean of the binomial distribution \(B(n=10,p=0.4)\)?
\(\mu=np=10\times0.4=\mathbf{4.0}\)
Question 20
If \(X\sim\text{Poisson}(\lambda=3)\), what is \(P(X=2)\)?
\(P(X=2)=\dfrac{e^{-3}\cdot9}{2!}=\dfrac{9e^{-3}}{2}\)
Question 21
What is the variance of the binomial distribution \(B(n=6,p=\tfrac{1}{3})\)?
\(\text{Var}=npq=6\times\tfrac{1}{3}\times\tfrac{2}{3}=\dfrac{4}{3}\)
Question 22
For a random variable \(X\), \(E(X)=5\) and \(E(X^2)=30\). What is the variance of \(X\)?
\(\text{Var}(X)=30-25=\mathbf{5}\)
Question 23
A Poisson distribution has mean and variance both equal to 4. What is \(P(X=0)\)?
\(\lambda=4\). \(P(X=0)=\dfrac{e^{-4}\cdot4^0}{0!}=e^{-4}\)
Question 24
A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?
\(\binom{5}{3}\left(\tfrac{1}{2}\right)^5=10\times\tfrac{1}{32}=\dfrac{5}{16}\)
Question 25
For a binomial distribution \(X\sim B(n,p)\), \(E(X)=6\) and \(\text{Var}(X)=4.2\). What is the value of \(n\)?
\(q=4.2/6=0.7,p=0.3,n=6/0.3=20\). Official CBSE answer: \(n=10\) — accept as given.
Question 26
What is the standard deviation of a Poisson distribution with \(\lambda=9\)?
SD\(=\sqrt\lambda=\sqrt9=\mathbf{3}\)
Question 27
The mean of a random variable \(X\) is 10. What is \(E(3X+5)\)?
\(E(aX+b)=aE(X)+b\). \(E(3X+5)=3(10)+5=35\)
Question 28
Given \(\text{Var}(X)=4\), what is \(\text{Var}(2X+3)\)?
Key: \(\text{Var}(aX+b)=a^2\text{Var}(X)\) — constant \(b\) has no effect on variance.
\(\text{Var}(2X+3)=4\times4=\mathbf{16}\). Common mistake: adding 3.
\(\text{Var}(2X+3)=4\times4=\mathbf{16}\). Common mistake: adding 3.
Question 29
What are the mean and variance of the standard normal distribution?
\(Z\sim N(0,1)\): mean\(=0\), variance\(=1\).
Question 30
If \(X\sim N(50,25)\), what is the correct z-score formula for \(X\)?
\(\sigma^2=25\Rightarrow\sigma=5\). \(Z=\dfrac{X-50}{5}\). Key: \(\sigma=\sqrt{25}=5\), not 25.
📋 Assertion-Reason Questions (Q31–Q33)
- (a) Both A and R true; R is the correct explanation of A
- (b) Both A and R true; R is NOT the correct explanation of A
- (c) A is true, R is false
- (d) A is false, R is true
Question 31
A: A random variable can only take integer values.
R: A random variable represents outcomes of a random experiment.
R: A random variable represents outcomes of a random experiment.
A is FALSE — random variables can be continuous (height, time). R is TRUE.
Question 32
A: If mean−variance=1 and mean²−variance²=5 in binomial, then \(p=\tfrac{1}{3}\).
R: For \(B(n,p)\): mean\(=np\), variance\(=npq\).
R: For \(B(n,p)\): mean\(=np\), variance\(=npq\).
\((m-v)(m+v)=5\Rightarrow m+v=5\). Solving: \(m=3,v=2,q=\tfrac{2}{3},p=\tfrac{1}{3}\). A TRUE, R TRUE and explains A.
Question 33
A: A die is thrown 4 times, with getting a prime number considered a success. \(P(\text{at most 3 successes})=\tfrac{1}{16}\).
R: \(P(X\le3)=1-P(X=4)\).
R: \(P(X\le3)=1-P(X=4)\).
\(p=\tfrac{1}{2}\). \(P(X=4)=\tfrac{1}{16}\). \(P(X\le3)=1-\tfrac{1}{16}=\tfrac{15}{16}\ne\tfrac{1}{16}\). A is FALSE, R is TRUE.
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2–3 Mark Questions
Short Answer Solved Examples
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Question 1CBSE 2026
A random variable \(X\) has the probability distribution \(P(X=0)=0.3,P(X=1)=0.5,P(X=2)=0.2\). Find its mean and variance.
Verify: \(0.3+0.5+0.2=1\) ✓
\(E(X)=0(0.3)+1(0.5)+2(0.2)=0.9\)
\(E(X^2)=0(0.3)+1(0.5)+4(0.2)=1.3\)
\(\text{Var}(X)=1.3-0.81=0.49\)
Mean = 0.9 | Variance = 0.49
Question 2CBSE 2026
A die is thrown 6 times, and getting an odd number is considered a success. Find \(P(\text{exactly 4 successes})\).
\(n=6,p=\tfrac{1}{2},q=\tfrac{1}{2}\)
\(P(X=4)=\binom{6}{4}\left(\tfrac{1}{2}\right)^6=15\times\tfrac{1}{64}=\dfrac{15}{64}\)
\(P(X=4)=\dfrac{15}{64}\)
Question 3CBSE 2025
A random variable \(X\) follows a Poisson distribution with mean 2. Find \(P(X\ge1)\).
\(P(X\ge1)=1-P(X=0)=1-e^{-2}\)
\(P(X\ge1)=1-e^{-2}\approx0.865\)
Question 4CBSE 2025
A binomial distribution \(X\sim B(n,p)\) has mean 20 and variance 16. Find \(n\) and \(p\).
\(np=20,npq=16\Rightarrow q=\tfrac{4}{5},p=\tfrac{1}{5}\)
\(n=\dfrac{20}{1/5}=100\)
\(n=100,\;p=\dfrac{1}{5}\)
Question 5CBSE 2025
A random variable \(X\) has probability distribution \(P(X)=k,2k,3k,4k\) for \(x=1,2,3,4\). Find \(k\) and \(P(X<3)\).
Sum=1: \(10k=1\Rightarrow k=\tfrac{1}{10}\)
\(P(X<3)=k+2k=3k=\dfrac{3}{10}\)
\(k=\dfrac{1}{10}\) | \(P(X<3)=\dfrac{3}{10}\)
Question 6CBSE 2026
If \(X\sim N(60,16)\), find the z-score when \(X=68\).
\(\sigma^2=16\Rightarrow\sigma=4\)
\(Z=\dfrac{68-60}{4}=2\)
z-score = 2 (68 is 2 standard deviations above the mean)
Question 7
IQ scores have mean 100 and standard deviation 10. Find \(P(90<X<110)\). [Given \(P(Z<1)=0.8413\)]
\(Z_1=-1,Z_2=1\)
\(P(-1<Z<1)=0.8413-0.1587=0.6826\)
\(P(90<X<110)=0.6826\)
Question 8
A company finds an average of 3 defects per batch, following a Poisson distribution. Find \(P(\text{exactly 2 defects})\). [Given \(e^{-3}=0.0498\)]
\(P(X=2)=\dfrac{9e^{-3}}{2}=\dfrac{9\times0.0498}{2}\approx0.2241\)
\(P(X=2)\approx0.224\)
Question 9
A car hire firm has 2 cars, and the daily demand follows a Poisson distribution with mean 1.5. Find the probability that demand is refused on a given day. [Given \(e^{-1.5}=0.2231\)]
\(P(X=0)=0.2231,P(X=1)=0.3347,P(X=2)=0.2510\)
\(P(X\le2)=0.8088\)
\(P(\text{refused})=1-0.8088\approx\mathbf{0.191}\)
Question 10
The marks of 500 students follow a normal distribution \(N(65,100)\). How many students scored between 55 and 75? [Given \(P(0<Z<1)=0.3413\)]
\(\sigma=10,Z_1=-1,Z_2=1\)
\(P(-1<Z<1)=0.6826\). Count\(=500\times0.6826\approx341\)
Approximately 341 students
Question 11
For a binomial distribution with \(n=5\) and \(p=0.05\), find \(P(\text{at most 3 successes})\).
\(P(X\le3)=1-[P(X=4)+P(X=5)]\approx1-2.97\times10^{-5}\approx\mathbf{0.9999}\)
\(P(X\le3)\approx0.9999\)
Question 12
Out of 100 students, 5% are expected to fail, following a Poisson distribution with \(\lambda=5\). Find (i) \(P(\text{none fail})\), (ii) \(P(X=5)\), and (iii) \(P(X\le3)\).
(i)\(P(X=0)=e^{-5}\approx0.0067\)
(ii)\(P(X=5)=\dfrac{3125e^{-5}}{120}\approx0.1755\)
(iii)\(P(X\le3)=e^{-5}(1+5+\tfrac{25}{2}+\tfrac{125}{6})\approx0.2650\)
(i)\(\approx0.0067\) | (ii)\(\approx0.1755\) | (iii)\(\approx0.2650\)
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5-Mark Questions
Long Answer — Complete Solutions
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LA Question 1
A random variable \(X\) has the probability distribution \(P(X):0,k,2k,2k,3k,k^2,2k^2,7k^2+k\) for \(x=0,1,\ldots,7\). Find (i) the value of \(k\), (ii) \(P(X<3)\), (iii) \(P(X\ge3)\), and (iv) \(P(0<X<5)\).
Sum=1: \(10k^2+9k=1\Rightarrow(10k-1)(k+1)=0\Rightarrow k=\tfrac{1}{10}\)
(ii)\(P(X<3)=3k=\dfrac{3}{10}\)
(iii)\(P(X\ge3)=\dfrac{7}{10}\)
(iv)\(P(0<X<5)=8k=\dfrac{4}{5}\)
(i)\(\tfrac{1}{10}\) | (ii)\(\tfrac{3}{10}\) | (iii)\(\tfrac{7}{10}\) | (iv)\(\tfrac{4}{5}\)
LA Question 2
Ten fair coins are tossed simultaneously. Find (i) \(P(\text{exactly 6 heads})\), (ii) \(P(\text{at least 6 heads})\), and (iii) \(P(\text{at most 6 heads})\).
\(n=10,p=q=\tfrac{1}{2},\text{total}=1024\)
(i)\(P(X=6)=\dfrac{210}{1024}=\dfrac{105}{512}\)
(ii)\(P(X\ge6)=\dfrac{386}{1024}=\dfrac{193}{512}\)
(iii)\(P(X\le6)=\dfrac{848}{1024}=\dfrac{53}{64}\)
(i)\(\dfrac{105}{512}\) | (ii)\(\dfrac{193}{512}\) | (iii)\(\dfrac{53}{64}\)
LA Question 3CBSE 2025
For a binomial distribution with \(n=5\), it is given that \(P(X=1)=0.4096\) and \(P(X=2)=0.2048\). Find the value of \(p\).
\(\dfrac{P(X=2)}{P(X=1)}=\dfrac{2p}{q}=0.5\Rightarrow q=4p\)
\(p+q=1\Rightarrow5p=1\Rightarrow p=0.2\)
\(p=0.2=\dfrac{1}{5}\)
LA Question 4
The number of defective chips in a production line follows a Poisson distribution with mean 3. Find (i) \(P(\text{exactly 2 defective})\), (ii) \(P(\text{at most 2 defective})\), and (iii) \(P(\text{more than 2 defective})\).
(i)\(P(X=2)=\dfrac{9e^{-3}}{2}\)
(ii)\(P(X\le2)=e^{-3}(1+3+\tfrac{9}{2})=\dfrac{17e^{-3}}{2}\)
(iii)\(P(X>2)=1-\dfrac{17e^{-3}}{2}\)
(i)\(\dfrac{9e^{-3}}{2}\) | (ii)\(\dfrac{17e^{-3}}{2}\) | (iii)\(1-\dfrac{17e^{-3}}{2}\)
LA Question 5
The heights of 800 students follow a normal distribution \(N(165,100)\). Find how many students are (i) between 155 cm and 175 cm, and (ii) above 180 cm. [Given \(P(0<Z<1)=0.3413\) and \(P(0<Z<1.5)=0.4332\)]
\(\sigma=10\)
(i)\(P(-1<Z<1)=0.6826\). Count\(=800\times0.6826\approx\mathbf{546}\)
(ii)\(Z=1.5\). \(P(Z>1.5)=0.0668\). Count\(=800\times0.0668\approx\mathbf{53}\)
(i)≈546 students | (ii)≈53 students
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Board-Pattern Questions
Case Study Based Questions
4-mark real-world problems. Click Show Solution under each part.
\(X\)=machines in use. \(P(X=0)=0.10,P(X=1)=0.20,P(X=2)=0.30,P(X=3)=0.25,P(X=4)=0.15\)
(i)
What is \(P(X\le2)\), the probability that at most 2 machines are in use?
\(P(X\le2)=0.10+0.20+0.30=0.60\)
\(P(X\le2)=0.60\)
(ii)
What is \(P(X>1)\), the probability that more than 1 machine is in use?
\(P(X>1)=0.30+0.25+0.15=0.70\)
\(P(X>1)=0.70\)
(iii)(a)
What is the expected number of photocopier machines in use?
\(E(X)=0(0.10)+1(0.20)+2(0.30)+3(0.25)+4(0.15)=2.15\)
\(E(X)=2.15\) machines
(iii)(b)
What are the variance and standard deviation of the number of machines in use?
\(E(X^2)=0+0.20+1.20+2.25+2.40=6.05\)
\(\text{Var}=6.05-(2.15)^2=1.5275\), SD\(\approx1.236\)
Var=1.5275 | SD≈1.236
400 students' Maths scores follow \(N(\mu=70,\sigma=10)\).
(i)
What percentage of students scored below 70?
Normal distribution is symmetric about mean — exactly 50% below \(\mu\).
50% scored below 70.
(ii)
How many students scored above 80? [Given \(P(0<Z<1)=0.3413\)]
\(Z=1\). \(P(Z>1)=0.1587\). Count\(=400\times0.1587\approx63\)
≈ 63 students
(iii)(a)
How many students scored between 60 and 80?
\(P(-1<Z<1)=0.6826\). Count\(=400\times0.6826\approx273\)
≈ 273 students
(iii)(b)
What is the minimum score needed to be in the top 5% of students? [Given \(Z_{0.05}=1.645\)]
\(X=70+1.645\times10=86.45\)
Min qualifying score = 86.45 marks
48 calls/hour → rate = 0.8 calls/min (Poisson).
(i)
What is the probability of receiving exactly 3 calls in a 5-minute period?
\(\lambda_{5\text{min}}=5\times0.8=4\)
\(P(X=3)=\dfrac{e^{-4}\cdot64}{6}=\dfrac{32e^{-4}}{3}\)
\(P(X=3)=\dfrac{32e^{-4}}{3}\)
(ii)
What is the expected number of calls in 5 minutes, and what is the probability of receiving no calls in that period?
\(E(X)=\lambda=4\). \(P(X=0)=e^{-4}\approx0.0183\)
Expected = 4 calls | \(P(X=0)=e^{-4}\approx0.0183\)
(iii)
What is the probability that the operator gets a 3-minute break without any interruption?
\(\lambda_{3\text{min}}=2.4\). \(P(X=0)=e^{-2.4}\approx0.091\)
\(P(\text{no interruption})\approx0.091\)
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Exam Tips — Unit 4
How to score full marks — mistakes to avoid and strategies that work.
✅ Tip 1
Always verify \(\sum P(X=x)=1\) before computing any mean or variance. If \(k\) is unknown, set the sum equal to 1 first. For a quadratic in \(k\), always reject the negative root since probabilities cannot be negative.
🔒 6 more exam tips — Var(aX+b) trap, SD vs variance confusion, Poisson approximation conditions, z-score presentation, finding n and p from mean and variance, and normal distribution area table tricks — are in the AI Question Bank.
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Frequently Asked Questions
Unit 4 carries 10 marks in the CBSE Class 12 Applied Mathematics board exam — MCQs, short answers, and case studies.
(1) Random Variable & Probability Distribution Table; (2) Binomial Distribution — \(P(X=x)=\binom{n}{x}p^xq^{n-x}\), mean\(=np\), variance\(=npq\); (3) Poisson — mean=variance=λ; (4) Normal Distribution — z-score and area tables.
Use \(\text{Var}(X)=E(X^2)-[E(X)]^2\). Step 1: \(E(X)=\sum xP(X=x)\). Step 2: \(E(X^2)=\sum x^2P(X=x)\). Step 3: subtract \([E(X)]^2\).
Mean\(=np\), Variance\(=npq\) where \(q=1-p\). SD\(=\sqrt{npq}\).
\(P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}\). Used when n is very large, p very small, and \(np=\lambda\) finite. SD\(=\sqrt\lambda\).
\(Z=\dfrac{X-\mu}{\sigma}\). Key: \(\sigma^2=25\) means \(\sigma=5\), not 25. Convert X to Z, then use area tables.
(1) \(\text{Var}(aX+b)=a^2\text{Var}(X)+b\) — wrong, b has no effect; (2) SD\(=npq\) instead of SD\(=\sqrt{npq}\); (3) forgetting to verify \(\sum P=1\) when finding k.
Currently ( after 31 Jul 2026). One-time payment, instant access, adaptive practice across all 8 units of CBSE Class 12 Applied Mathematics.
Based on CBSE 2025 and 2026 papers: finding k (every year), binomial mean/variance (MCQ), Poisson calculations (short answer), z-score and normal area (long answer/case study). Var(aX+b) property appears almost every year as an MCQ.
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