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Business Mathematics and Statistics · Class 12 Commerce

Ch 7Probability Distributions — Class 12 Business Mathematics and Statistics, concept-first.

In the previous chapter you met the idea of a random variable — a variable whose value is determined by the outcome of a chance experiment (the number of heads in three coin tosses, the number of defective pieces in a sample, the daily sales of a shop).

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Key concepts

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Binomial Distribution

Imagine you flip a fair coin 10 times. You want to know the probability of getting exactly 6 heads. This is the kind of question the binomial distribution answers — it models the number of "successes" in a fixed number o…

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CA Foundation

2 concepts · 19 appearances

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Introduction: From Random Variables to Probability Distributions

In the previous chapter you met the idea of a random variable — a variable whose value is determined by the outcome of a chance experiment (the number of heads in three coin tosses, the number of defe…

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Binomial Distribution: Conditions and Probability Mass Function

The binomial distribution is the theoretical distribution that applies whenever an experiment consists of a fixed number of Bernoulli trials — trials that satisfy all four of these conditions:

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Mean, Variance and a Worked Illustration of the Binomial Distribution

Two summary numbers describe a binomial distribution completely, without needing to list every individual probability:

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Poisson Distribution: The Law of Rare Events

The binomial distribution assumes a known, fixed (the number of trials) and a not-too-small probability of success .

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Normal Distribution: Shape and Properties

The binomial and Poisson distributions both describe a random variable that can only take whole-number values — a batch can have 2 defectives or 3, never 2.4.

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The Standard Normal Variate and Reading Areas from the Normal Table

Every normal distribution — whatever its mean and standard deviation — can be converted into the same reference distribution, called the standard normal distribution, which has mean and standard devia…

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Choosing the Right Distribution for a Business Problem

With three theoretical distributions now available, the practical skill this chapter builds toward is recognising, from the wording of a business problem, which one applies. A simple decision guide:

Exercises

+Show 7 questions7 questions
  1. Q4In a batch inspection, the probability that any single item is defective is 0.1. If 6 items are picked independently, find the probability t…Free
  2. Q7The number of calls arriving at a call centre in a given minute follows a Poisson distribution with a mean of 4 calls per minute. Find the p…Free
  3. Q8In a large batch of 100 items, the probability that any one item is defective is 0.02. Using the Poisson approximation to the binomial distr…Free
  4. Q10For the same shop (mean ₹5,000, standard deviation ₹800), find the probability that daily sales lie between ₹4,200 and ₹5,800. (Area between…Preview
  5. Q11The marks scored by a large batch of students in an examination are normally distributed with a mean of 60 and a standard deviation of 10. F…Preview
  6. Q12State, with reasons, which probability distribution — Binomial, Poisson or Normal — is most appropriate for each of the following situations…Preview
  7. Q13State the mean and variance of the Binomial, Poisson and Normal distributions, and explain the special relationship that holds between the m…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 24 questions24 questions
  1. Q1A manufacturer produces switches and experiences that $2$ percent switches are defective. The probability that in a box of $50$ switches, th…Preview
  2. Q2The mean of Binomial distribution is $20$ and variance is $16$. Find $"p"$ and $"n"$.Preview
  3. Q3Write down any three chief characteristics of normal probability curve.Preview
  4. Q4(a) The mean score of $500$ students for an examination is $40$ and S.D is $25$. Determine the limit of the marks of the central $60\%$ of t…Preview
  5. Q5A manufacturer produces switches and experiences that 2 percent switches are defective. The probability that in a box of 50 switches, there…Preview
  6. Q6In a binomial distribution, the probability of success is twice as that of failure. Then out of 4 trials, the probability of no success is _…Preview
  7. Q7If the mean of binomials distribution is 20 and standard deviation is 4, then find the value of 'n'.Preview
  8. Q8Write the conditions for which the poisson distribution is a limiting case of binomial distribution.Preview
  9. Q9Normal distribution was invented by : (a) Gauss (b) Laplace (c) James Bernoulli (d) De-MoivrePreview
  10. Q10In a parametric distribution the mean is equal to variance is : (a) normal (b) poisson (c) binomial (d) all of the abovePreview
  11. Q11Define Binomial distribution.Preview
  12. Q12In a book of 520 pages, 390 typo-graphical errors occur. Assuming Poisson law for the number of errors per page, find the probability that a…Preview
  13. Q13In turning out certain toys in a manufacturing company, the average number of defective is 1%. The probability that in the sample of 100 toy…Preview
  14. Q14Using the Standard normal table the sum of the probabilities to the right of $z=2.18$ and to the left of $z=-1.75$ is : (a) 0.0146 (b) 0.485…Preview
  15. Q15The mean of a binomial distribution is 5 and standard deviation is 2. Determine the distribution.Preview
  16. Q16A probability that a student get a degree is 0.4. Determine the probability that out of 5 students (i) one will be graduate (ii) atleast one…Preview
  17. Q17If the parameters of a binomial distribution $B(n,p)$ mean $=4$ and variance $=\dfrac{4}{3}$, the probability, $P(X\ge 5)$ is equal to : (a)…Preview
  18. Q18In a binomial distribution, the probability of success is twice as that of failure, then out of 4 trials, the probability of no success is :…Preview
  19. Q19The mean of Binomial distribution is 20 and Standard deviation is 4. Find the parameters of the distribution.Preview
  20. Q20A fair coin is tossed 7 times. Find the probability that exactly 2 heads occur.Preview
  21. Q21If $X \sim N(5, 25)$ then the standard normal variate Z will be : (a) $Z = \frac{X - 5}{5}$ (b) $Z = \frac{X - 25}{5}$ (c) $Z = \frac{X - 25…Preview
  22. Q22Which of the following cannot generate a Poisson distribution ? (a) The number of bacteria found in a cubic foot of soil (b) The number of t…Preview
  23. Q23Five fair coins are tossed simultaneously. Find the probability of getting exactly 3 heads.Preview
  24. Q24X is a normally distributed variable with mean $\mu = 30$ and standard deviation $\sigma = 4$. Find $P(30 < X < 35)$. | $Z$ | $1.25$ | $0$ |…Preview

More questions

+Show 6 questions6 questions
  1. Example 1A fair coin is tossed 4 times. Find the probability of getting exactly 2 heads.Free
  2. Example 2In the same coin-tossing experiment (4 tosses of a fair coin), find the probability of getting at least 3 heads.Free
  3. Example 3For a binomial distribution with n = 5 and p = 0.4, list the complete probability distribution, and verify that the mean and variance obtain…Preview
  4. Example 5A factory finds that, on average, 3 bulbs are defective per day's production. Assuming the number of defective bulbs per day follows a Poiss…Preview
  5. Example 6The average number of accidents at a certain factory is 2 per day. Assuming a Poisson distribution, find (i) the probability of no accident…Preview
  6. Example 9The daily sales of a retail shop are normally distributed with a mean of ₹5,000 and a standard deviation of ₹800. Find the probability that…Preview