Applied Mathematics · Class 12 Commerce
Ch 6Probability Distribution — Class 12 Applied Mathematics, concept-first.
This concept map shows how the chapter fits together. A probability distribution may be discrete — a random variable with its mean (expectation) and variance, and the Binomial and Poisson distributions — or continuous — the Normal distribution, its standard form, the Z-score and the Z-test.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Random Variable Definition
Imagine you're about to toss a coin three times. Before you toss, the outcome is uncertain — it could be HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT. But you're not really interested in the exact sequence.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the chapter fits together. A probability distribution may be discrete — a random variable with its mean (expectation) and variance, and the Binomial and Poisson distribution…
Introduction
Many experiments in probability have outcomes that aren't numbers by themselves — the colour of a sweater drawn at random, the face shown on a tossed coin, and so on.
Discrete and Continuous Random Variables
A random variable can behave in one of two fundamentally different ways, depending on the kind of values it can take.
Probability Distribution of Discrete Random Variable
Once a discrete random variable has been defined on a sample space, the natural next step is to ask: how likely is each of its possible values? Consider rolling two dice and letting be the sum of the…
Mathematical Expectation of Discrete Probability Distribution
The mean, as a measure of central tendency, gives a rough sense of the middle or average value taken by a random variable across an experiment.
+−Worked Examplesi2 questions
- Example 3A coin is tossed twice and outcomes are recorded. Prepare the probability distribution table for random variable X which represents the numb…Free
- Example 4In a manufacturing unit inspection, from a lot of 20 baskets which include 6 defectives, a sample of 2 baskets is drawn at random without re…Preview
Variance of Discrete Probability Distribution
While the mean locates the centre of a random variable's values, the variance measures how spread out those values tend to be around that centre.
+−Worked Examplesi2 questions
- Example 5A class XII has 20 students whose marks (out of 30) are 14, 17, 25, 14, 21, 17, 17, 19, 18, 26, 18, 17, 17, 26, 19, 21, 21, 25, 14 and 19 ye…Free
- Example 6Let X denote the number of hours a person watches television during a randomly selected day. The probability that X can take the values $x_i…Preview
Binomial Distribution
Many experiments boil down to just two possible outcomes — a coin toss lands heads or tails, an answer is correct or incorrect, a component is defective or not.
+−Worked Examplesi6 questions
- Example 7Prepare the Binomial distribution $B\left(4, \frac{2}{3}\right)$Free
- Example 8If a fair coin is tossed 9 times, find the probability of a) exactly five tails b) At least five tails c) At most five tailsFree
- Example 9In a manufacturing unit inspection, from a lot of 20 baskets which include 6 defectives, a sample of 2 baskets is drawn at random with repla…Preview
- Example 10The probability that Rohit will hit a shooting target is $\frac{2}{3}$. While preparing for an international shooting competition, Rohit aim…Preview
- Example 11Sonal and Anannya are playing a game by throwing a die alternatively till one of them gets a '1' and wins the game. Find their respective pr…Preview
- Example 12A die is thrown again and again until three 5s' are obtained. Find the probability of obtaining the third 5 in the seventh throw of the diePreview
Poisson Distribution
Some experiments count how many times an event occurs over a fixed span of time, distance, area or volume — the number of car sales at a showroom in a day, or the number of customers arriving at a res…
+−Worked Examplesi4 questions
- Example 13As the story goes, the Prussian soldiers monitored 10 cavalry corps over a period of 20 years. The annual number of recorded deaths due to h…Free
- Example 14A traffic engineer records the number of bicycle riders that use a particular cycle track. He records that an average of 3.2 bicycle riders…Free
- Example 15A particular river near a small-town floods and overflows twice in every 10-years on an average. Assuming that the Poisson distribution is a…Preview
- Example 16For a Poisson distribution model, if arrival rate of passengers at an airport is recorded as 30 per hour on a given day. Find: a) The expect…Preview
Normal Distribution
All the distributions discussed so far describe a discrete random variable, where every possible value can be listed and assigned its own probability.
Standard Normal Distribution
A normal distribution is fully described by just two parameters — its mean and standard deviation . In practice, though, different data sets rarely share the same mean and standard deviation: one set…
Z-Score of Normal Distribution
A normal distribution with mean and standard deviation is called the standard normal distribution, and the variable following it is called a standard normal variate; the distribution is also referred…
+−Worked Examplesi2 questions
- Example 17Calculate Z-Score for a normal distribution of length of 7 rare species of Indian butterfly that you have in your garden | Butterfly | 1 | 2…Free
- Example 18Given that mean of a normal variate X is 12 and standard deviation is 4, then find: a) Find the Z-Score of data point 20 b) The data point i…Preview
Z-Test for Normal Distribution
When a claim is made about a population — for instance, that a new drug is effective, or that a process meets some target — a hypothesis test is used to judge how likely that claim is to be true, base…
+−Worked Examplesi2 questions
- Example 19In a district, exam scores of 300 student of class XII are recorded at the end of the session. a) Ramesh scored 800 marks in total out of 10…Free
- Example 20Given that the scores of a set of candidates on an IQ test are normally distributed. If the IQ test has a mean of 100 and a standard deviati…Preview
Quick Recapitulation
This chapter introduced random variables and the main probability distributions used in applied statistics. Its key ideas and formulae are gathered below.
Check Your Progress
+−Check Your Progress24 questions
- Q1State which of the following are not the probability distributions of a random variable. Give reasons for your answer a. | X | -1 | 0 | 1 |…Free
- Q2A lady's bag contains 2 black and 1 red pens. One pen is drawn at random and then put back in the box after noting its colour. The process i…Free
- Q3What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face?Free
- Q4Raheem tossed a fair coin 10 times, find the probability of (i) exactly six heads (ii) at least six heads (iii) at most six heads.Preview
- Q5Find the probability of getting 5 exactly twice in 7 throws of a fair die.Preview
- Q6Let X denote the number of hours a class XII student studies during a randomly selected school day. The probability that X can take the valu…Preview
- Q7How many times must Sumit toss a fair coin so that the probability of getting at least one head is more than 90%?Preview
- Q8A pair of dice is thrown and the random variable X represents the sum of the numbers that appear on the two dice. Calculate the mathematical…Preview
- Q9Find the variance of a bernoulli random variable whose probability of success is 0.6.Preview
- Q10If the mean and variance of a binomial distribution are $4/3$ and $8/9$ resp. find $P(x=1)$Preview
- Q11What is the expected value of number of tails on a throw of a fair coin?Preview
- Q12A customer care company receives an average of 4.5 calls every 5 minutes. If each customer executive can handle one of these calls over the…Preview
- Q13A statistician records the number of trucks approaching a particular intersection to analyze the flow of traffic. He observes that on an ave…Preview
- Q14A computer disk manufacturer tests disk quality on random basis before approving it. The approval is based on the number of errors in a test…Preview
- Q15In a Poisson distribution, if mean is 2, what is the variance?Preview
- Q16It is given that 3% defective electric bulb are manufactured by a company. Using Poisson distribution, find the probability of 100 bulbs wil…Preview
- Q17The mortality rate for a certain disease is 0.007. Using Poisson distribution, calculate the probability for 2 deaths in a group of 400 peop…Preview
- Q18An ice-cream parlour receives a customer at an average rate of 4 per minute. If the number of customers received by the parlour follows a Po…Preview
- Q19Using Z-Table, Calculate a) $P(Z < 1.20)$ b) $P(Z \le 1.20)$ c) $P(-0.5 \le Z \le 1.0)$ d) $P(-1.0 \le Z \le 1.0)$Preview
- Q20A company conducted an IQ test for randomly select 50 employees. Volunteer A scored 74 out of the possible 120 points. If the average IQ tes…Preview
- Q21An average ceiling fan manufactured by the Jagdeep Corporation lasts 300 days with a standard deviation of 50 days. Assuming that the ceilin…Preview
- Q22In a survey of daily travel time (in minutes) of students to reach their school was recorded as follows: 26, 33, 65, 28, 34, 55, 25, 44, 50,…Preview
- Q23In an examination, 2000 students appeared and the mean of the normal distribution of marks is 30 with standard deviation as 6.25. Find out h…Preview
- Q24In a normal distribution, 31% of the articles are under 45 and 8% are over 64. Calculate the mean and standard deviation of the distributionPreview