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Statistics · Class 12 Commerce

Ch 5Probability — Class 12 Statistics, concept-first.

Every problem in this chapter begins with a random experiment — an experiment whose outcome cannot be predicted with certainty in advance, but whose full set of possible outcomes is known beforehand.

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Key concepts

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Random Experiment, Sample Space and Algebra of Events

A random experiment has an unpredictable outcome but a known set of possible outcomes, the sample space ; an event is any subset of . Basic vocabulary: trial, outcome, favourable outcomes, equally likely outcomes.

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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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Random Experiment, Sample Space and Algebra of Events

Every problem in this chapter begins with a random experiment — an experiment whose outcome cannot be predicted with certainty in advance, but whose full set of possible outcomes is known beforehand.

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The Three Approaches to Probability: Classical, Relative Frequency and Axiomatic

There are three distinct, standard ways of assigning a probability to an event, and the Gujarat board Std 12 Statistics syllabus expects a student to be able to state, apply, and compare all three.

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The Addition Theorem of Probability

The Addition Theorem answers the question: what is the probability that AT LEAST ONE of two events occurs, i.e. ?

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Conditional Probability and the Multiplication Theorem

Conditional probability asks: given that event has already occurred, what is the probability that also occurs? It is written ('probability of given ') and defined as:

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Independent Events

Two events and are called independent if the occurrence of one has absolutely no effect on the probability of the other — knowing that has occurred gives no new information about , so and .

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Bayes' Theorem (Introduction)

Bayes' Theorem answers a distinctive kind of question: given that an OUTCOME has already been observed, what is the probability it came from one particular CAUSE among several possible causes? This re…

Exercises

+Show 6 questions6 questions
  1. Q8Define the following terms with one example each: (i) Random experiment, (ii) Sample space, (iii) Mutually exclusive events, (iv) Exhaustive…Free
  2. Q9Explain the classical, relative-frequency and axiomatic approaches to probability. What is the main limitation of the classical approach?Free
  3. Q10In a class of 60 students, 25 study Economics, 20 study Statistics, and 10 study both subjects. A student is selected at random from the cla…Preview
  4. Q11If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, find P(A | B) and P(B | A).Preview
  5. Q12If A and B are independent events with P(A) = 0.3 and P(B) = 0.4, then P(A ∩ B) equals: (a) 0.7 (b) 0.12 (c) 0.1 (d) cannot be determinedPreview
  6. Q13A factory has three machines A, B and C producing 25%, 35% and 40% of its total output respectively. Their defective rates are 5%, 4% and 2%…Preview

Sample & Board Papers

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

+Show 31 questions31 questions
  1. Q1What is the value of $P(A \cap A')$ for events $A$ and $A'$? (a) $1$ (b) $0$ (c) $0.5$ (d) between $0$ and $1$Preview
  2. Q2Write the sample space of a random experiement of throwing one balanced die and a balanced coin simultaneously.Preview
  3. Q33 bulbs are defective in a box of 10 bulbs. 2 bulbs are randomly selected from this box. These bulbs are fixed in two bulb holders installed…Preview
  4. Q4A number is selected from the natural number 1 to 100. Find the probability of the event that the selected number is a multiple of 3 or 5.Preview
  5. Q5(A) Find the probability of getting 'D' in the first place in all possible arrangements of each and every letter of the word 'DHYAN'. (B) Tw…Preview
  6. Q6What is the other name of the classical definition of probability? (a) Mathematical Definition (b) Axiomatic Definition (c) Statistical Defi…Preview
  7. Q7If $P(A/B) = P(A)$ and $P(B/A) = P(B)$ then what type of events are $A$ and $B$? (a) Independent events (b) Complementary events (c) Certain…Preview
  8. Q8State the formula for the probability of occurrence of at least one event out of three events A, B and C.Preview
  9. Q9Find the probability of getting D in the first place in all possible arrangements of each and every letter of the word DHYAN.Preview
  10. Q10If two balanced coins are tossed, then find the probability of (1) getting one head and one tail (2) getting at least one head.Preview
  11. Q11A number of selected from the natural number 1 to 100. Find the probability of the event that the selected number is a multiple of 3 or 5.Preview
  12. Q12For three mutually exclusive and exhaustive events $A$, $B$ and $C$ in the sample space of a random experiment $2P(A) = 3P(B) = 4P(C)$. Find…Preview
  13. Q13What is the total number of sample points in the sample space formed by throwing three six-faced balanced dice simultaneously? (a) (A) $6^2$…Preview
  14. Q14For two events $A$ and $B$ in a sample space, $A \cap B = \phi$ and $A \cup B = U$, then state the values of $P(A \cap B)$ and $P(A \cup B)$…Preview
  15. Q15One number is randomly selected from the natural numbers 1 to 100. Find the probability that the selected number is either a single digit nu…Preview
  16. Q16The $P(A') = \frac{7}{25}$, $P(B/A) = \frac{5}{12}$ and $P(A/B) = \frac{1}{2}$ for two events $A$ and $B$ in the sample space of a random ex…Preview
  17. Q17(A) Find the probability of getting P in the first place in all possible arrangements of each and every letter of the word PUJAN. OR (B) Fin…Preview
  18. Q18Which random experiment from the following random experiments has an infinite sample space? (a) Throwing two dice (b) Selecting two employee…Preview
  19. Q19What is the probability of having 5 Mondays in the month of February in a year which is not a leap year? (a) $0$ (b) $\frac{1}{7}$ (c) $\fra…Preview
  20. Q20Define mutually exclusive events.Preview
  21. Q21If two balanced coins are tossed, then find the probability of : (i) getting one head and one tail. (ii) getting at least one head.Preview
  22. Q22State the characteristics of random experiment.Preview
  23. Q23If $P(B) = \frac{3}{5}$ and $P(A' \cap B) = \frac{1}{2}$, for two events $A$ and $B$, find $P\left(\frac{A}{B}\right)$ and $P(A' \cup B')$.Preview
  24. Q24One number is randomly selected from the natural number 1 to 100. Find the probability that the number selected is either a single digit num…Preview
  25. Q25Which of the following options is true for any event A of the sample space? (a) $P(A) < 0$ (b) $0 \le P(A) \ge 1$ (c) $0 \le P(A) \le 1$ (d)…Preview
  26. Q26What is the total number of sample points in the sample space formed by throwing three six-faced balanced dice simultaneously? (a) $6^2$ (b)…Preview
  27. Q27Arrange $P(A \cup B), P(A), P(A \cap B), 0, P(A) + P(B)$ in the ascending order.Preview
  28. Q28State the law of addition of probability for two events $A$ and $B$. Write the law of addition of probability if these two events are mutual…Preview
  29. Q29Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that (1) Both the cards are of different colours. (2) Both t…Preview
  30. Q30For two events A and B in the sample space of a random experiment $P(A') = 0.3,\ P(B) = 0.6$ and $P(A \cup B) = 0.83$, find $P(A \cap B')$ a…Preview
  31. Q31There are two children in a family. If the first child is a girl then find the probability that both the children in the family are girls.Preview

More questions

+Show 7 questions7 questions
  1. Example 1Two dice are thrown together. Write the sample space and find the number of sample points. Hence find the probability of getting a sum of 7…Free
  2. Example 2A drawing pin was tossed 500 times, and it landed 'point up' 316 times. Using the relative-frequency (empirical) approach, estimate the prob…Free
  3. Example 3A card is drawn from a well-shuffled pack of 52 playing cards. Find the probability that the card drawn is a king or a heart.Free
  4. Example 4A bag contains 4 red, 5 white and 6 blue balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is red or…Preview
  5. Example 5A bag contains 5 red and 3 black balls. Two balls are drawn one after another, WITHOUT replacement. Find the probability that both balls dra…Preview
  6. Example 6A coin is tossed and a die is thrown simultaneously. Find the probability of getting a head on the coin and a 6 on the die.Preview
  7. Example 7Bag I contains 4 white and 6 black balls; Bag II contains 6 white and 4 black balls. A bag is selected at random and a ball is drawn from it…Preview