Mathematics and Statistics · Class 11 Commerce
Ch 16Probability — Class 11 Mathematics and Statistics, concept-first.
Probability is the branch of mathematics that measures how likely an uncertain event is to happen. Before we can measure that likelihood we must describe the situation precisely, and that begins with the idea of a random experiment.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Classical Definition of Probability
For a finite sample space of equally likely outcomes, , the ratio of favourable to total outcomes. Always , with , , and .
Most relevant Q&A
- Three coins are tossed together. Find the probability of getting (i) exactly two heads, (ii) at least one head.Free
- A die is rolled once. Find the probability of getting (i) a number greater than $4$, (ii) a number that is not a multiple of $3$.Free
- Two dice are rolled. Find the probability that the sum of the numbers on the two dice is (i) $7$, (ii) at least $10$.Free
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.
Random Experiments and Sample Space
Probability is the branch of mathematics that measures how likely an uncertain event is to happen. Before we can measure that likelihood we must describe the situation precisely, and that begins with…
Events and Their Types
An event is any subset of the sample space — that is, a collection of one or more outcomes that we are interested in. Events are usually written with capital letters .
Classical (Mathematical) Definition of Probability
When all the outcomes of a random experiment are equally likely (no outcome is favoured over another — a fair coin, a fair die, a well-shuffled pack), the probability of an event is defined by simple…
Addition Theorem of Probability
The addition theorem gives the probability that at least one of two events occurs — the probability of the union .
Conditional Probability
Often we learn that one event has already happened and want the probability of another event in the light of that information.
Multiplication Theorem and Independent Events
Rearranging the definition of conditional probability gives a rule for the probability that both events occur — the probability of the intersection .
Exercises
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- Q9Three coins are tossed together. Find the probability of getting (i) exactly two heads, (ii) at least one head.Free
- Q10A card is drawn from a pack of $52$. Are the events $A$ = "the card is a king" and $B$ = "the card is a heart" independent? Justify using pr…Free
- Q11The probability that a student passes Mathematics is $0.8$ and passes Statistics is $0.7$. If the probability of passing at least one subjec…Preview
- Q12A problem is given to three students $A$, $B$, $C$ whose chances of solving it are $\dfrac{1}{2}$, $\dfrac{1}{3}$ and $\dfrac{1}{4}$ respect…Preview
- Q13A box contains $6$ good and $4$ defective bulbs. Two bulbs are drawn at random without replacement. Find the probability that (i) both are d…Preview
- Q14If $P(A)=\dfrac{2}{5}$, $P(B)=\dfrac{1}{3}$ and $P(A\cap B)=\dfrac{1}{5}$, find $P(A\mid B)$ and $P(B\mid A)$.Preview
More questions
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- Example 1Two coins are tossed simultaneously. Write the sample space $S$ and state $n(S)$. Also write the event $A$: "at least one head appears" and…Free
- Example 2A die is rolled once. Find the probability of getting (i) a number greater than $4$, (ii) a number that is not a multiple of $3$.Free
- Example 3Two dice are rolled. Find the probability that the sum of the numbers on the two dice is (i) $7$, (ii) at least $10$.Free
- Example 4A card is drawn at random from a well-shuffled pack of $52$ cards. Find the probability that it is a face card or a spade.Preview
- Example 5For two events $A$ and $B$, $P(A)=0.5$, $P(B)=0.4$ and $P(A\cup B)=0.7$. Find (i) $P(A\cap B)$, (ii) $P(A'\cap B')$.Preview
- Example 6A number is chosen at random from $1$ to $20$. Given that the chosen number is even, find the probability that it is a multiple of $4$.Preview
- Example 7A bag contains $5$ red and $4$ black balls. Two balls are drawn one after another **without replacement**. Find the probability that both ar…Preview
- Example 8$A$ and $B$ are independent events with $P(A)=0.3$ and $P(B)=0.6$. Find (i) $P(A\cap B)$, (ii) $P(A\cup B)$, (iii) $P(A\mid B)$.Preview