Q.A die is thrown. If $E$ is the event 'the number appearing is a multiple of 3' and $F$ be the event 'the number appearing is even' then find whether $E$ and $F$ are independent?

Showing the 12 most recent of 55 on this concept.

1. CBSE 2025Set 65/4/11 markMCQ
   Q.Assertion (A) : If A and B are two events such that $P(A \cap B) = 0$, then A and B are independent events. Reason (R) : Two events are independent if the occurrence of one does not affect the occurrence of the other. (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) Assertion (A) is false, but Reason (R) is true.

   ›Reveal solutionSolution

   The assertion confuses mutually exclusive events ($P(A \cap B) = 0$) with independent events ($P(A \cap B) = P(A) \cdot P(B)$); the reason correctly defines independence. The answer is (D).

   The heart of this question lies in distinguishing two fundamentally different relationships between events: mutual exclusivity and independence. These concepts are often confused because both involve restrictions on how events relate, but they describe opposite scenarios.

   Understanding Independence

   The reason (R) gives the correct intuitive definition: two events are independent when the occurrence of one does not affect the probability of the other occurring. Mathematically, events $A$ and $B$ are independent if and only if:

   $P(A \cap B) = P(A) \cdot P(B)$

   Equivalently, independence means $P(A \mid B) = P(A)$ (when $P(B) > 0$) and $P(B \mid A) = P(B)$ (when $P(A) > 0$). The events "don't care" about each other.

   Why the Assertion Fails

   Now let's examine what $P(A \cap B) = 0$ actually tells us.

   1. What does $P(A \cap B) = 0$ mean?

   This condition says that events $A$ and $B$ cannot occur simultaneously—they are mutually exclusive or disjoint. If one happens, the other cannot.

   2. Testing for independence

   For $A$ and $B$ to be independent, we need $P(A \cap B) = P(A) \cdot P(B)$. If $P(A \cap B) = 0$, then independence requires:

   $0 = P(A) \cdot P(B)$

   This equation holds only if at least one of $P(A)$ or $P(B)$ equals zero—meaning at least one event is impossible.

   3. The typical case

   If both $A$ and $B$ have positive probabilities (both are possible events), then $P(A) \cdot P(B) > 0$. But we're told $P(A \cap B) = 0$. This means:

   $P(A \cap B) = 0 \neq P(A) \cdot P(B)$

   The events are not independent. In fact, they are maximally dependent: knowing one occurred tells you with certainty that the other did not.

   Watch out

   Mutually exclusive events with positive probabilities are always dependent, not independent. If $A$ happens, it completely rules out $B$—that's maximum dependence, not independence!

   4. A concrete example

   Consider rolling a fair die. Let $A$ = "rolling a 2" and $B$ = "rolling a 5."

   • $P(A) = \frac{1}{6}$, $P(B) = \frac{1}{6}$
   • $P(A \cap B) = 0$ (can't roll both 2 and 5 simultaneously)
   • For independence we'd need: $P(A \cap B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$

   But $0 \neq \frac{1}{36}$, so these events are not independent despite having $P(A \cap B) = 0$.

   Tip

   Independence and mutual exclusivity are nearly opposite concepts. Independent events can happen together (with probability equal to the product of their individual probabilities), while mutually exclusive events cannot happen together at all.

   Evaluating the Options

   • Assertion (A): Claims $P(A \cap B) = 0$ implies independence. This is false (except in the trivial case where at least one event is impossible).

   • Reason (R): Correctly defines independence. This is true.

   Since the assertion is false but the reason is true, the correct answer is (D).

   ✓Final answer

   The correct option is (D): Assertion (A) is false, but Reason (R) is true.

2. CBSE 2026Set 65/3/11 markMCQ
   Q.If $E$ and $F$ are two independent events such that $P(E) = \frac{3}{10}$, $P(E \cup F) = \frac{1}{2}$, then $P(E|F) - P(F|E)$ is equal to: (A) $\frac{2}{7}$ (B) $\frac{3}{35}$ (C) $\frac{1}{70}$ (D) $\frac{1}{7}$

   ›Reveal solutionSolution

   We use the property of independent events, $P(E \cap F) = P(E)P(F)$, along with the union formula to first find $P(F)$. Then, we use the fact that for independent events, $P(E|F) = P(E)$ and $P(F|E) = P(F)$, to calculate the required difference. The final result is $\boxed{\frac{1}{70}}$.

   The core of this problem lies in understanding how the concept of "independent events" simplifies probability calculations, especially when dealing with unions and conditional probabilities.

   When two events, $E$ and $F$, are independent, it means that the occurrence of one event does not affect the probability of the other event occurring. This has two crucial implications:

   1. Intersection Probability: The probability of both $E$ and $F$ happening, $P(E \cap F)$, is simply the product of their individual probabilities: $P(E \cap F) = P(E)P(F)$.
   2. Conditional Probability: The probability of $E$ happening given that $F$ has already happened, $P(E|F)$, is just the probability of $E$, because $F$'s occurrence doesn't change $E$'s likelihood. So, $P(E|F) = P(E)$. Similarly, $P(F|E) = P(F)$.

   We are given $P(E)$, $P(E \cup F)$, and that $E$ and $F$ are independent. Our strategy will be to first use the formula for the union of events, combined with the independence property, to find $P(F)$. Once we have $P(F)$, we can directly use the independence property to find $P(E|F)$ and $P(F|E)$, and then calculate their difference.

   1. Find $P(F)$ using the union formula and independence.

      The general formula for the probability of the union of two events is:

      $P(E \cup F) = P(E) + P(F) - P(E \cap F)$

      Since $E$ and $F$ are independent, we can substitute $P(E \cap F)$ with $P(E)P(F)$:

      $P(E \cup F) = P(E) + P(F) - P(E)P(F)$

      Now, substitute the given values: $P(E) = \frac{3}{10}$ and $P(E \cup F) = \frac{1}{2}$.

      $\frac{1}{2} = \frac{3}{10} + P(F) - \frac{3}{10} P(F)$

      To solve for $P(F)$, group the terms involving $P(F)$:

      $\frac{1}{2} = \frac{3}{10} + P(F) \left(1 - \frac{3}{10}\right)$

      $\frac{1}{2} = \frac{3}{10} + P(F) \left(\frac{7}{10}\right)$

      Subtract $\frac{3}{10}$ from both sides:

      $\frac{1}{2} - \frac{3}{10} = P(F) \left(\frac{7}{10}\right)$

      To subtract the fractions on the left, find a common denominator, which is $10$:

      $\frac{5}{10} - \frac{3}{10} = P(F) \left(\frac{7}{10}\right)$

      $\frac{2}{10} = P(F) \left(\frac{7}{10}\right)$

      Now, isolate $P(F)$ by multiplying both sides by $\frac{10}{7}$:

      $P(F) = \frac{2}{10} \times \frac{10}{7}$

      $P(F) = \frac{2}{7}$

   2. Calculate $P(E|F)$.

      For independent events $E$ and $F$, the probability of $E$ occurring given that $F$ has occurred is simply the probability of $E$. This is because $F$'s occurrence provides no new information about $E$.

      $P(E|F) = P(E)$

      We are given $P(E) = \frac{3}{10}$.

      So, $P(E|F) = \frac{3}{10}$.

   3. Calculate $P(F|E)$.

      Similarly, for independent events $E$ and $F$, the probability of $F$ occurring given that $E$ has occurred is simply the probability of $F$.

      $P(F|E) = P(F)$

      From Step 1, we found $P(F) = \frac{2}{7}$.

      So, $P(F|E) = \frac{2}{7}$.

   4. Calculate $P(E|F) - P(F|E)$.

      Now, substitute the values we found:

      $P(E|F) - P(F|E) = \frac{3}{10} - \frac{2}{7}$

      To subtract these fractions, find a common denominator, which is $70$:

      $\frac{3}{10} - \frac{2}{7} = \frac{3 \times 7}{10 \times 7} - \frac{2 \times 10}{7 \times 10}$

      $= \frac{21}{70} - \frac{20}{70}$

      $= \frac{1}{70}$

   ✓Final answer

   The value of $P(E|F) - P(F|E)$ is $\boxed{\frac{1}{70}}$.

3. CBSE 2026Set V11 markMCQ
   Q.The probability of obtaining an even prime number on each die when a pair of dice is rolled

   (a) $\frac{1}{36}$

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

   (c) $\frac{1}{18}$

   (d) $\frac{1}{4}$

   ›Reveal solutionSolution

   The even prime is $2$; $P(2\text{ on each of two dice})=\tfrac{1}{6}\cdot\tfrac{1}{6}=\tfrac{1}{36}$; answer (a).

   The only even prime number is $2$. For one die, $P(\text{show }2)=\dfrac{1}{6}$. The two dice are independent, so

   $P(\text{even prime on each})=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}.$

   ✓Final answer

   (a) $\frac{1}{36}$

4. CBSE 2026Set V11 markMCQ
   Q.If $A$ and $B$ are independent events with $P(A)=0.3$ and $P(B)=0.4$ then $P(A\cap B)$

   (a) $1.2$

   (b) $0.12$

   (c) $0.7$

   (d) $\frac{3}{4}$

   ›Reveal solutionSolution

   Independence gives $P(A\cap B)=P(A)P(B)=0.12$; answer (b).

   For independent events $A$ and $B$,

   $P(A\cap B)=P(A)\cdot P(B)=0.3\times0.4=0.12.$

   (Note (a) $1.2$ is impossible since a probability cannot exceed $1$.)

   ✓Final answer

   (b) $0.12$

5. CBSE 2026Set A1 markMCQ
   Q.If $A$, $B$ and $C$ are three independent events then $P(ABC) =$

   (a) $P(A) + P(B) + P(C)$

   (b) $P(A) - P(B) - P(C)$

   (c) $P(A) \cdot P(B) \cdot P(C)$

   (d) None of these

   ›Reveal solutionSolution

   For independent events, $P(A\cap B\cap C) = P(A)P(B)P(C)$.

   By definition, events $A$, $B$, $C$ are (mutually) independent when the probability of their joint occurrence equals the product of their individual probabilities:

   $P(ABC) = P(A)\cdot P(B)\cdot P(C).$

   Adding probabilities (option a) applies to mutually exclusive unions, not to intersections of independent events.

   ✓Final answer

   (c) $P(A)\cdot P(B)\cdot P(C)$.

6. CBSE 2026Set ANNUAL1 markMCQ
   Q.If A and B are independent events and $P(A) = 0.3$ and $P(B) = 0.4$, then the value of $P(A \cup B)$ will be

   (a) $0.58$

   (b) $0.70$

   (c) $0.12$

   (d) $0.10$

   ›Reveal solutionSolution

   For independent events, $P(A\cap B)=P(A)P(B)$, and the addition rule gives $P(A\cup B)$.

   $P(A\cap B) = 0.3\times0.4=0.12$ (independence).

   $P(A\cup B)=P(A)+P(B)-P(A\cap B) = 0.3+0.4-0.12=0.58$.

   ✓Final answer

   The correct option is (a) $0.58$.

7. CBSE 2026Set ANNUAL1 markMCQ
   Q.The probability of obtaining an even prime number on each dice, when a pair of dice is rolled, is:

   (a) 0

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

   (c) $\dfrac{1}{12}$

   (d) $\dfrac{1}{36}$

   ›Reveal solutionSolution

   The only even prime number is 2, so we need a 2 on each die.

   Among $\{1,2,3,4,5,6\}$, the only even prime is $2$.

   $P(\text{2 on one die})=\dfrac{1}{6}$

   Since the two dice are independent, $P(\text{2 on both dice})=\dfrac{1}{6}\times\dfrac{1}{6}=\dfrac{1}{36}$

   ✓Final answer

   Option (d): $\dfrac{1}{36}$

8. CBSE 2026Set ANNUAL1 markMCQ
   Q.Ajay and Meera are contesting for two vacancies in a company. Probability of selection of Ajay is 7/9 and that of Meera is 4/7. What is the probability that both will be rejected?

   (a) 61/63

   (b) 6/63

   (c) 41/63

   (d) 28/63

   ›Reveal solutionSolution

   The rejection probabilities of each candidate are complements of their selection probabilities; since the two events are independent, multiply them.

   $P(\text{Ajay selected}) = \dfrac79 \implies P(\text{Ajay rejected}) = 1-\dfrac79 = \dfrac29$

   $P(\text{Meera selected}) = \dfrac47 \implies P(\text{Meera rejected}) = 1-\dfrac47 = \dfrac37$

   Since selection of Ajay and Meera are independent events:

   $P(\text{both rejected}) = \frac29 \times \frac37 = \frac{6}{63}$

   ✓Final answer

   The probability that both are rejected is $\dfrac{6}{63}$ (option b).

9. CBSE 2026Set ANNUAL1 markMCQ
   Q.Assertion (A): Two events A and B are such that P(A) = 1/4, P(B) = 1/2 and P(A∩B) = 1/8 then two events A and B are independent. Reason (R): Two events are independent if the probability of occurrence of one does not affect the probability of occurrence of other and P(A∩B) = P(A) + P(B) − P(A∪B)

   (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

   The independence check in A is numerically correct, but R states the wrong criterion — it gives the addition-rule identity (true for ANY two events), not the actual independence condition $P(A\cap B)=P(A)\cdot P(B)$.

   Checking Assertion (A): Independence requires $P(A\cap B) = P(A)\cdot P(B)$.

   $P(A)\cdot P(B) = \frac14 \times \frac12 = \frac18,$

   which equals the given $P(A\cap B)=1/8$. So A and B are independent — A is true.

   Checking Reason (R): R correctly describes independence in words ("occurrence of one does not affect the other"), but then states the test as

   $P(A\cap B) = P(A)+P(B)-P(A\cup B).$

   This is just the general addition-rule identity, rearranged — it holds for every pair of events, independent or not, and is NOT the criterion that establishes independence (that criterion is $P(A\cap B)=P(A)P(B)$, which R never states). So as a stated reason for independence, R is false/incorrect.

   ✓Final answer

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

10. CBSE 2026Set ANNUAL1 markMCQ
    Q.Let $E$ and $F$ be events with $P(E) = \dfrac{1}{3}$, $P(F) = \dfrac{1}{2}$ and $P(E \cap F) = \dfrac{1}{6}$. Then

    (a) $E$ and $F$ are independent events

    (b) $E$ and $F$ are mutually exclusive events

    (c) $E$ and $F$ are disjoint events

    (d) None of the above

    ›Reveal solutionSolution

    Two events are independent exactly when $P(E\cap F)=P(E)\cdot P(F)$; check whether the given numbers satisfy this.

    Given $P(E)=\dfrac13$, $P(F)=\dfrac12$, $P(E\cap F)=\dfrac16$.

    Test for independence:

    $P(E)\cdot P(F) = \frac13\times\frac12 = \frac16$

    This equals the given $P(E\cap F)=\dfrac16$. Since $P(E\cap F)=P(E)P(F)$, $E$ and $F$ are independent.

    Rule out the other options: mutually exclusive/disjoint events would require $P(E\cap F)=0$, but here $P(E\cap F)=\dfrac16\ne0$, so $E$ and $F$ are not mutually exclusive (and "disjoint" means the same thing as mutually exclusive).

    ✓Final answer

    (a) $E$ and $F$ are independent events

11. CBSE 2026Set ANNUAL1 mark
    Q.If A and B are two independent events with P(A) = 1/2 and P(B) = 1/3, then find P(A ∪ B).

    ›Reveal solutionSolution

    For independent events, $P(A\cap B)=P(A)P(B)$; then apply the addition rule.

    $P(A\cup B) = P(A)+P(B)-P(A)P(B) = \dfrac12+\dfrac13-\dfrac12\cdot\dfrac13 = \dfrac12+\dfrac13-\dfrac16 = \dfrac{3+2-1}{6}=\dfrac{4}{6}=\dfrac23$

    ✓Final answer

    $P(A\cup B) = 2/3$.

12. CBSE 2026Set ANNUAL1 mark
    Q.Write the answer in one word/sentence: If $E$ and $F$ are independent events then write the value of $P(E \cap F)$.

    ›Reveal solutionSolution

    For independent events, the probability of the intersection is the product.

    By definition, events $E$ and $F$ are independent iff $P(E\cap F)=P(E)P(F)$.

    ✓Final answer

    $P(E\cap F)=P(E)\,P(F)$.