Q.Case Study - 1 Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below: $P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3 \\ 2kx, & \text{for } x = 4, 5, 6 \\ 0, & \text{otherwise} \end{cases}$ where $x$ denotes the number of hours. Based on the above information, answer the following questions:

Showing the 12 most recent of 88 on this concept.

1. CBSE 2026Set 65/1/11 markMCQ
   Q.Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that the number appearing on the die is odd is $\dfrac{2}{3}$. Reason (R): For any two events $A$ and $B$, $P(A|B) = \dfrac{P(A \cup B)}{P(B)}$. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

   ›Reveal solutionSolution

   The assertion is true: given the outcome is odd, the probability it is a prime is $\frac{2}{3}$. The reason states the correct conditional probability formula. Since the reason directly justifies the calculation in the assertion, both are true and the reason is the correct explanation.

   Concept first — Conditional probability asks: If we already know that event $B$ has occurred, what is the probability that event $A$ also occurs? The sample space shrinks from all possible outcomes to just those in $B$. The formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ is the precise way to compute this reduced probability.

   Here, the die is unbiased, so each face $\{1,2,3,4,5,6\}$ has probability $\frac{1}{6}$. The assertion involves two events:

   • $A$: the number is prime. On a die, the primes are $2,3,5$.
   • $B$: the number is odd. The odd numbers are $1,3,5$.

   The condition "given that the number is odd" means we restrict attention to $B = \{1,3,5\}$. Among these three equally likely outcomes, the primes are $3$ and $5$ — that's two out of three. So the conditional probability is $\frac{2}{3}$.

   Now let's verify step by step using the formula in Reason (R).

   1. Define the events precisely.

      $A = \{2,3,5\}$, $B = \{1,3,5\}$.

      The sample space $S = \{1,2,3,4,5,6\}$.

   2. Compute $P(B)$.

      $B$ has 3 outcomes, each with probability $\frac{1}{6}$, so $P(B) = \frac{3}{6} = \frac{1}{2}$.

   3. Compute $P(A \cap B)$.

      $A \cap B$ = numbers that are both prime and odd = $\{3,5\}$. That's 2 outcomes, so $P(A \cap B) = \frac{2}{6} = \frac{1}{3}$.

   4. Apply the formula from Reason (R).

   $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{1/3}{1/2} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3}.$

   This matches the assertion exactly. …

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2. CBSE 2026Set V11 mark
   Q.Choose from $[0, 3, -1, 2, -2, 1]$. If $F$ is an event of a sample space $S$ then $P(S|F)=$ ____.

   ›Reveal solutionSolution

   Since $S\cap F=F$, the conditional probability $P(S\mid F)=1$.

   By the definition of conditional probability (with $P(F)\neq0$),

   $P(S\mid F)=\frac{P(S\cap F)}{P(F)}.$ …

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3. CBSE 2026Set CX1 markMCQ
   Q.If $3P(A)=P(B)=\dfrac{5}{13}$ and $P(A/B)=\dfrac{2}{5}$, then $P(A\cup B)$ will be:

   (a) $\dfrac{20}{39}$

   (b) $\dfrac{16}{39}$

   (c) $\dfrac{11}{39}$

   (d) $\dfrac{14}{39}$

   ›Reveal solutionSolution

   Using $P(A\cap B)=P(A/B)P(B)$ and the addition rule gives $P(A\cup B)=\dfrac{14}{39}$ — option (d).

   Given: $3P(A)=P(B)=\dfrac{5}{13}$ and $P(A/B)=\dfrac{2}{5}$.

   So $P(B)=\dfrac{5}{13}$ and $P(A)=\dfrac{1}{3}\cdot\dfrac{5}{13}=\dfrac{5}{39}$.

   Intersection (multiplication rule):

   $P(A\cap B)=P(A/B)\,P(B)=\dfrac{2}{5}\cdot\dfrac{5}{13}=\dfrac{2}{13}.$

   …

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4. CBSE 2026Set A1 markMCQ
   Q.$P(A) = \frac{7}{13},\ P(B) = \frac{9}{13},\ P(A \cap B) = \frac{4}{13} \Rightarrow P(A/B) =$

   (a) $\frac{4}{9}$

   (b) $\frac{4}{7}$

   (c) $\frac{12}{13}$

   (d) $\frac{1}{6}$

   ›Reveal solutionSolution

   $P(A|B) = \frac{4}{9}$.

   Use the conditional-probability definition:

   $P(A|B) = \frac{P(A\cap B)}{P(B)}.$

   Substitute the given values: …

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5. CBSE 2026Set ANNUAL1 markMCQ
   Q.If $P(B)=0.5$ and $P(A\cap B)=0.32$, then write the value of $P(A\mid B)$.

   (a) $\frac{15}{23}$

   (b) $\frac{16}{25}$

   (c) $\frac{16}{27}$

   (d) $\frac{16}{23}$

   ›Reveal solutionSolution

   By the definition of conditional probability, $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{16}{25}$.

   The conditional probability of $A$ given $B$ is defined as

   $P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B) eq0$

   …

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6. CBSE 2026Set ANNUAL1 mark
   Q.A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?

   ›Reveal solutionSolution

   List the equally likely outcomes for two children, restrict to those with at least one boy, then find the fraction that are both boys.

   Sample space $=\{BB, BG, GB, GG\}$, each equally likely.

   Given at least one boy: reduced sample space $=\{BB,BG,GB\}$ (3 outcomes).

   …

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7. CBSE 2026Set ANNUAL1 markMCQ
   Q.If $P(A) = 0.8$, $P(B) = 0.5$ and $P\left(\dfrac{B}{A}\right) = 0.4$ then $P(A \cap B) =$

   (a) 0.8

   (b) 0.5

   (c) 0.32

   (d) 0.4

   ›Reveal solutionSolution

   Use the multiplication rule of conditional probability: $P(A\cap B) = P(B\mid A)\cdot P(A)$.

   Given $P(A)=0.8$, $P(B\mid A)=0.4$.

   $P(A\cap B) = P(B\mid A)\cdot P(A) = 0.4 \times 0.8 = 0.32$.

   …

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8. CBSE 2026Set ANNUAL1 markMCQ
   Q.If $P(A) = \dfrac{3}{10}$, $P(B) = \dfrac{2}{5}$ and $P(A \cup B) = \dfrac{3}{5}$, then $P(B/A)$ is:

   (a) $\dfrac{1}{4}$

   (b) $\dfrac{1}{3}$

   (c) $\dfrac{5}{12}$

   (d) $\dfrac{7}{12}$

   ›Reveal solutionSolution

   Find $P(A\cap B)$ from the addition rule, then use the conditional probability formula.

   Given $P(A)=\dfrac{3}{10}$, $P(B)=\dfrac{2}{5}$, $P(A\cup B)=\dfrac{3}{5}$.

   …

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9. CBSE 2026Set ANNUAL1 mark
   Q.A doctor is to visit a patient. From past experience it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\dfrac{3}{10}, \dfrac{1}{5}, \dfrac{1}{10}$ and $\dfrac{2}{5}$. The probabilities that he will be late are $\dfrac{1}{4}, \dfrac{1}{3}$ and $\dfrac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. Probability that he will not be late, when he comes by other means of transport.

   ›Reveal solutionSolution

   The problem statement itself states that if the doctor comes by other means, he will never be late.

   Let $E_1,E_2,E_3,E_4$ denote the events that the doctor comes by train, bus, scooter, or other means, with $P(E_1)=\dfrac{3}{10}$, $P(E_2)=\dfrac{1}{5}$, $P(E_3)=\dfrac{1}{10}$, $P(E_4)=\dfrac{2}{5}$ (these sum to 1, confirming they form a complete set of cases).

   …

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10. CBSE 2026Set ANNUAL1 markMCQ
    Q.If P(A) = 1/2, P(B) = 3/8 and P(A∪B) = 27/40 then P(A/B) is equal to:

    (a) 2/5

    (b) 8/15

    (c) 2/3

    (d) 5/8

    ›Reveal solutionSolution

    First find $P(A\cap B)$ using the addition rule, then apply the conditional probability formula $P(A/B)=\frac{P(A\cap B)}{P(B)}$.

    Given $P(A) = \frac12$, $P(B)=\frac38$, $P(A\cup B) = \frac{27}{40}$.

    Using the addition rule $P(A\cup B) = P(A)+P(B)-P(A\cap B)$:

    $P(A\cap B) = P(A)+P(B)-P(A\cup B) = \frac{20}{40}+\frac{15}{40}-\frac{27}{40} = \frac{8}{40} = \frac15$

    …

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11. CBSE 2026Set ANNUAL1 markMCQ
    Q.Assertion (A): If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4 then P(A∩B) = 0.32 Reason (R): Conditional Probability of 'B' when event A has occurred is given by P(B/A) = P(A∩B) / P(A)

    (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

    (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

    (c) Assertion (A) is true but Reason (R) is false.

    (d) Both Assertion (A) and Reason (R) are false.

    ›Reveal solutionSolution

    Applying the conditional-probability formula in R directly to the given numbers reproduces the value claimed in A.

    Reason (R): $P(B/A) = \dfrac{P(A\cap B)}{P(A)}$ is the standard definition of conditional probability — true.

    Checking Assertion (A): Given $P(A)=0.8,\ P(B/A)=0.4$. Rearranging R's formula:

    $P(A\cap B) = P(B/A)\times P(A) = 0.4 \times 0.8 = 0.32.$ …

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12. CBSE 2026Set ANNUAL1 markMCQ
    Q.If $P(A) = \dfrac{1}{2}$ and $P(B) = 0$, then $P(A|B)$ is

    (a) 0

    (b) $\dfrac{1}{2}$

    (c) 1

    (d) Not defined

    ›Reveal solutionSolution

    Conditional probability is defined as $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$, which requires dividing by $P(B)$ — impossible when $P(B)=0$.

    Given $P(A)=\dfrac12$, $P(B)=0$.

    By definition:

    $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$

    …

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