Mathematics · Class 11 Science
Ch 14Probability — Class 11 Mathematics, concept-first.
In everyday language, an experiment often means a scientific test carried out under controlled conditions. In probability theory the word is used more broadly: an experiment is any action or process that produces some observable result.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Addition Theorem of Probability
For any two events and of a sample space , : the overlap is subtracted once because it was counted twice when and were added separately.
Most relevant Q&A
- In a survey of $100$ students, $60$ like Physics, $45$ like Chemistry, and $25$ like both subjects. If a student is selected at random, find…Free
- If $A$ and $B$ are events such that $P(A) = 0.5$, $P(B) = 0.35$ and $P(A \cup B) = 0.7$, determine whether $A$ and $B$ are mutually exclusiv…Preview
- In a class, the probability that a randomly chosen student likes Mathematics is $0.5$, the probability that the student likes Science is $0.…Preview
- If $P(A) = 0.42$, $P(B) = 0.48$ and $P(A \cap B) = 0.16$, find $P(A \cup B)$ and $P(A' \cap B')$ (the probability that neither $A$ nor $B$ o…Preview
- A card is drawn from a well-shuffled deck of $52$ playing cards. Find the probability that the card is a king or a queen.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Random Experiments and Outcomes
In everyday language, an experiment often means a scientific test carried out under controlled conditions.
Sample Space (Set Representation)
The sample space of a random experiment is the set of all possible outcomes of that experiment. It is usually denoted by , and each individual outcome is called an element, or a sample point, of .
Events as Subsets of the Sample Space
Once the sample space of a random experiment is fixed, an event is defined formally as any subset of . This is exactly the set-theoretic language from the Sets chapter, now applied to probability.
Types of Events — Exhaustive and Mutually Exclusive Events
Two properties of a collection of events matter enormously when their probabilities are combined: whether the events can occur simultaneously, and whether they cover every possible outcome between the…
Axiomatic (Set-Theoretic) Probability
Earlier classes may have introduced probability informally as for experiments with equally likely outcomes. The axiomatic approach, due to the mathematician A. N.
Probability of 'Not' an Event
A very useful consequence of the three axioms is a formula for the probability of the complementary event ("not ") in terms of alone — derived directly from the axioms, not assumed separately.
Probability of 'And'/'Or' Events
Axiom 3 gives only when and are mutually exclusive. When and can occur together — i.e. — simply adding and double-counts the overlap. The addition theorem (general addition rule) corrects for this.
Summary
This chapter built probability on a rigorous, set-theoretic foundation.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1In single throw of two dice, the probability of obtaining 'a total of 8' is (a) 8/36 (b) 3/36 (c) 9/36 (d) 5/36.Preview
- Q2If P(A) = 2/3, P(B) = 1/2, P(A∩B) = 1/6, then find the value of P(A∩B') and P(A∪B).Preview
- Q3A bag contains 5 white and 4 black balls. If 3 balls are drawn at random, find the probability that at least two of them are white.Preview
More questions
30 Q+−Show 3 questionsHide questions3 questions
- Q28In a survey of $100$ students, $60$ like Physics, $45$ like Chemistry, and $25$ like both subjects. If a student is selected at random, find…Free
- Q29Two dice are thrown together, and the sum of the numbers appearing is observed. Let $A =$ 'the sum is a multiple of $3$'. Find $P(A)$ and $P…Preview
- Q30If $A$ and $B$ are events such that $P(A) = 0.5$, $P(B) = 0.35$ and $P(A \cup B) = 0.7$, determine whether $A$ and $B$ are mutually exclusiv…Preview
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- Example 1A coin is tossed twice. Write the sample space of this random experiment.Free
- Example 2A die is thrown once. Let $A$ be the event 'the number is even' and $B$ be the event 'the number is greater than $4$'. Write $A$, $B$, $A \c…Free
- Example 3Two dice are thrown together. Write the event $E =$ 'the sum of the numbers is $8$' as a subset of the sample space, and find $n(E)$.Free
- Example 4A card is drawn from a well-shuffled deck of $52$ playing cards. Let $A =$ 'the card is a king' and $B =$ 'the card is a heart'. Are $A$ and…Preview
- Example 5Three coins are tossed together. Write the sample space, and list the outcomes in the event 'at least two heads appear'.Preview
- Example 6If $P(A) = 0.4$, $P(B) = 0.5$, and $A$, $B$ are mutually exclusive events, find $P(A \cup B)$ and $P(A \cap B)$.Preview
- Example 7If $P(A) = 0.6$ for an event $A$, find $P(A')$, the probability of 'not $A$'.Preview
- Example 8In a class, the probability that a randomly chosen student likes Mathematics is $0.5$, the probability that the student likes Science is $0.…Preview
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- Q9A coin and a die are thrown together (simultaneously). Write the sample space of this experiment, and find $n(S)$.Preview
- Q10A die is thrown twice. Write the sample space in set-builder form, and state $n(S)$.Preview
- Q11A bag contains $3$ red and $2$ white balls. Two balls are drawn one after another without replacement, and only the colour is noted each tim…Preview
- Q12A die is thrown once. Let $A =$ 'the number is a multiple of $3$' and $B =$ 'the number is even'. Write $A$, $B$ and $A \cap B$ as subsets o…Preview
- Q13A card is drawn from a well-shuffled deck of $52$ cards. Let $E =$ 'the card is a spade' and $F =$ 'the card is a face card (a king, queen o…Preview
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- Q14A die is thrown once. Let $A = \{1, 3, 5\}$, $B = \{2, 4, 6\}$, $C = \{1, 2\}$. Determine, giving reasons, which of the pairs $(A,B)$, $(A,C…Free
- Q15In the experiment of throwing a die once, let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. Show that $A$ and $B$ are both mutually exclusive and…Free
- Q16Two dice are thrown together. Let $A =$ 'the sum of the two numbers is $7$' and $B =$ 'the sum of the two numbers is $11$'. Are $A$ and $B$…Preview
- Q17A coin is tossed twice. Let $A =$ 'at least one head appears' and $B =$ 'no head appears'. Show that $A$ and $B$ are mutually exclusive and…Preview
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- Q18A random experiment has sample space $S = \{a, b, c, d\}$ with all outcomes equally likely. Using the axioms of probability, find $P(\{a\})$…Free
- Q19If $P(A) = 0.3$, $P(B) = 0.45$ and $A \cap B = \phi$, verify that this assignment is consistent with the axioms of probability, and find $P(…Free
- Q20A probability function assigns $P(1) = 0.1$, $P(2) = 0.2$, $P(3) = 0.3$, $P(4) = 0.4$ to the outcomes of $S = \{1, 2, 3, 4\}$. Verify that t…Preview
- Q21Can the following be a valid probability assignment for $S = \{1, 2, 3\}$: $P(1) = 0.5$, $P(2) = 0.6$, $P(3) = -0.1$? Justify your answer us…Preview
- Q22A sample space $S$ has $10$ equally likely outcomes. Find the probability of an event $E$ with $n(E) = 4$, using the axiomatic definition of…Preview
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- Q23If $P(A) = 0.35$ for an event $A$, find $P(\text{not } A)$.Free
- Q24If $P(A) = \dfrac{7}{13}$ for an event $A$, find $P(A')$.Free
- Q25If $P(A) = 0.42$, $P(B) = 0.48$ and $P(A \cap B) = 0.16$, find $P(A \cup B)$ and $P(A' \cap B')$ (the probability that neither $A$ nor $B$ o…Preview
- Q26A card is drawn from a well-shuffled deck of $52$ playing cards. Find the probability that the card is a king or a queen.Preview
- Q27Two dice are thrown together. Find the probability that the sum of the numbers is either $7$ or $11$.Preview