Q.If $P(A) = \frac{7}{13}$, $P(B) = \frac{9}{13}$ and $P(A \cap B) = \frac{4}{13}$, evaluate $P(A|B)$.

Showing the 12 most recent of 45 on this concept.

1. KCET 2022Set C-41 markMCQ
   Q.If A and B are two events such that $P(A) = \dfrac{1}{2}$, $P(B) = \dfrac{1}{3}$ and $P(A|B) = \dfrac{1}{4}$, then $P(A' \cap B')$ is (A) $3/16$ (B) $1/12$ (C) $3/4$ (D) $1/4$

   ›Reveal solutionSolution

   Get $P(A\cap B)$ from the conditional probability, use the addition rule for $P(A\cup B)$, then apply De Morgan: $P(A'\cap B')=1-P(A\cup B)$.

   Step 1 — Recover the joint probability from the conditional.

   Conditional probability is defined by $P(A|B)=\dfrac{P(A\cap B)}{P(B)}$ — it re-scales the probability of $A$ to the reduced sample space $B$. Rearranging gives the multiplication rule:

   $P(A\cap B)=P(A|B)\,P(B)=\frac14\times\frac13=\frac1{12}.$

   Step 2 — Addition rule for the union.

   $P(A\cup B)=P(A)+P(B)-P(A\cap B)=\frac12+\frac13-\frac1{12}.$

   Taking LCM $12$:

   $P(A\cup B)=\frac{6}{12}+\frac{4}{12}-\frac{1}{12}=\frac{9}{12}=\frac34.$

   Step 3 — De Morgan's law.

   The event "neither $A$ nor $B$ occurs" is exactly the complement of "$A$ or $B$ occurs":

   $A'\cap B'=(A\cup B)'.$

   Hence

   $P(A'\cap B')=1-P(A\cup B)=1-\frac34=\frac14.$

   ✓Final answer

   The correct option is (D) — $1/4$.

   ANSWER: D

2. KCET 2020Set A-11 markMCQ
   Q.If A and B are two events such that $P(A) = \frac{1}{3}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{6}$, then $P(A'/B)$ is (A) $\frac{2}{3}$ (B) $\frac{1}{3}$ (C) $\frac{1}{2}$ (D) $\frac{1}{12}$

   ›Reveal solutionSolution

   Use $P(A'|B) = 1 - P(A|B)$ (equivalently $\frac{P(B)-P(A\cap B)}{P(B)}$), which gives $\frac{2}{3}$.

   Step 1 — The definition of conditional probability.

   For $P(B) > 0$,

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

   Conditioning on $B$ means we restrict the sample space to $B$ and ask what fraction of $B$ lies outside $A$.

   Step 2 — Split $B$ into the part inside $A$ and the part outside $A$.

   The events $A \cap B$ and $A' \cap B$ are disjoint and their union is exactly $B$:

   $P(B) = P(A \cap B) + P(A' \cap B)$

   $\Rightarrow P(A' \cap B) = P(B) - P(A \cap B) = \frac{1}{2} - \frac{1}{6} = \frac{3-1}{6} = \frac{2}{6} = \frac{1}{3}$

   Step 3 — Divide by $P(B)$.

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

   Step 4 — Cross-check with the complement rule.

   First find $P(A|B) = \dfrac{P(A\cap B)}{P(B)} = \dfrac{1/6}{1/2} = \dfrac{1}{3}$. Since $A$ and $A'$ partition the space, $P(A|B) + P(A'|B) = 1$, so

   $P(A'|B) = 1 - \frac{1}{3} = \frac{2}{3}$

   Both routes agree. (Note $P(A) = \tfrac13$ was not even needed — a useful reminder that a conditional probability given $B$ depends only on how $B$ is split.)

   ✓Final answer

   The correct option is (A) — $\dfrac{2}{3}$.

   ANSWER: A

3. COMEDK 2025Set 2025-E1 markMCQ
   Q.If A and B are two events such that $P(\bar{A})=0.3, P(B)=0.4, P(A \cap \bar{B})=0.5$, then find the value of $P(B / A \cup \bar{B})$ (A) 0.33 (B) 0.7 (C) 0.8 (D) 0.25

   ›Reveal solutionSolution

   The problem asks for a conditional probability $P(B \mid A \cup \bar{B})$. Using the definition of conditional probability and the given probabilities, we compute $P(B \cap (A \cup \bar{B}))$ and $P(A \cup \bar{B})$, then divide. The final value is $0.25$, so the correct option is (D).

   We are given:

   • $P(\bar{A}) = 0.3$, so $P(A) = 1 - 0.3 = 0.7$.
   • $P(B) = 0.4$.
   • $P(A \cap \bar{B}) = 0.5$.

   We need $P(B \mid A \cup \bar{B})$, which is the probability that $B$ occurs given that $A$ or $\bar{B}$ (or both) occurs.

   Concept and intuition:

   Conditional probability $P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$. Here $X = B$ and $Y = A \cup \bar{B}$. So we need the probability that both $B$ and $(A \cup \bar{B})$ happen, divided by the probability that $A \cup \bar{B}$ happens. The key is to simplify the intersection $B \cap (A \cup \bar{B})$ using set algebra — it often collapses to something simpler.

   1. Simplify the numerator $P(B \cap (A \cup \bar{B}))$

      Using the distributive law:

      $B \cap (A \cup \bar{B}) = (B \cap A) \cup (B \cap \bar{B})$

      But $B \cap \bar{B} = \emptyset$, so this is just $B \cap A$.

      Hence $P(B \cap (A \cup \bar{B})) = P(A \cap B)$.

   2. Find $P(A \cap B)$

      We know $P(A) = 0.7$ and $P(A \cap \bar{B}) = 0.5$.

      Since $A = (A \cap B) \cup (A \cap \bar{B})$ and these are disjoint,

      $P(A) = P(A \cap B) + P(A \cap \bar{B})$

      $\Rightarrow 0.7 = P(A \cap B) + 0.5$

      $\Rightarrow P(A \cap B) = 0.2$.

   3. Find the denominator $P(A \cup \bar{B})$

      Use the inclusion-exclusion principle:

      $P(A \cup \bar{B}) = P(A) + P(\bar{B}) - P(A \cap \bar{B})$

      We have $P(\bar{B}) = 1 - P(B) = 1 - 0.4 = 0.6$.

      So $P(A \cup \bar{B}) = 0.7 + 0.6 - 0.5 = 0.8$.

   4. Compute the conditional probability

      $P(B \mid A \cup \bar{B}) = \frac{P(A \cap B)}{P(A \cup \bar{B})} = \frac{0.2}{0.8} = 0.25$.

   Tip

   A common mistake is to forget that $B \cap \bar{B} = \emptyset$ and try to compute $P(B \cap (A \cup \bar{B}))$ directly — but the distributive trick saves time and avoids errors.

   Watch out

   Do not confuse $P(A \cup \bar{B})$ with $P(A) + P(\bar{B})$ — they are not disjoint because $A$ and $\bar{B}$ can overlap (and indeed do, since $P(A \cap \bar{B}) = 0.5$).

   ✓Final answer

   The correct option is (D).

   ANSWER: D

4. KCET 2025Set A-11 markMCQ
   Q.If A and B are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct? (A) $P(A|B) = \dfrac{P(B)}{P(A)}$ (B) $P(A|B) < P(A)$ (C) $P(A|B) \geq P(A)$ (D) $P(A) = P(B)$

   ›Reveal solutionSolution

   Use $A \subset B \Rightarrow A \cap B = A$, then note that dividing $P(A)$ by $P(B) \le 1$ cannot make it smaller.

   Step 1 — Simplify the intersection.

   If every element of $A$ lies in $B$, then $A \cap B = A$. Hence

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

   Step 2 — Compare with $P(A)$.

   Every probability satisfies $0 < P(B) \le 1$ (we are told $P(B) \ne 0$). Therefore $\dfrac{1}{P(B)} \ge 1$, and multiplying the non-negative number $P(A)$ by a factor $\ge 1$ gives

   $P(A\mid B) = P(A)\cdot\frac{1}{P(B)} \;\ge\; P(A).$

   Equality holds exactly when $P(B) = 1$ (or when $P(A) = 0$).

   Intuition: conditioning on $B$ throws away all outcomes outside $B$ — but none of $A$ lies outside $B$. So $A$'s share of the shrunken sample space can only grow.

   Step 3 — Eliminate.

   • (A) $P(A|B) = P(B)/P(A)$ — wrong; the ratio is upside down (and can exceed 1).
   • (B) $P(A|B) < P(A)$ — the inequality points the wrong way.
   • (D) $P(A) = P(B)$ — not implied; $A$ can be a strictly smaller subset, e.g. rolling a die with $A = \{2\}$, $B = \{2,4,6\}$: $P(A) = 1/6$, $P(B) = 1/2$, and $P(A|B) = 1/3 \ge 1/6$. ✓ consistent only with (C).

   ✓Final answer

   The correct option is (C) — $P(A \mid B) \ge P(A)$.

   ANSWER: C

5. COMEDK 2025Set 2025-A1 markMCQ
   Q.If for two events $A$ and $B, P(A-B)=\frac{1}{5}$ and $P(A)=\frac{3}{5}$ then $P(B / A)=$ (A) $\frac{2}{3}$ (B) $\frac{1}{2}$ (C) $\frac{3}{5}$ (D) $\frac{2}{5}$

   ›Reveal solutionSolution

   The key is to interpret $P(A-B)$ as $P(A \cap B^c)$ and use the definition of conditional probability. The result is $P(B/A) = \frac{2}{3}$, so option (A) is correct.

   We are asked for $P(B/A)$, the probability of $B$ given $A$. The definition is

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

   We know $P(A) = \frac{3}{5}$, so we need $P(A \cap B)$. The given $P(A-B) = \frac{1}{5}$ is the key: $A-B$ means “$A$ and not $B$,” i.e., $A \cap B^c$.

   1. Relate $P(A-B)$ to $P(A \cap B)$ Since $A$ is the union of the disjoint parts “$A$ and $B$” and “$A$ and not $B$,” we have

   $P(A) = P(A \cap B) + P(A \cap B^c).$

   Here $P(A \cap B^c) = P(A-B) = \frac{1}{5}$ and $P(A) = \frac{3}{5}$.

   2. Solve for $P(A \cap B)$

   $\frac{3}{5} = P(A \cap B) + \frac{1}{5} \quad\Rightarrow\quad P(A \cap B) = \frac{3}{5} - \frac{1}{5} = \frac{2}{5}.$

   3. Apply the conditional probability formula

   $P(B/A) = \frac{P(A \cap B)}{P(A)} = \frac{2/5}{3/5} = \frac{2}{3}.$

   Watch out

   A common mistake is to confuse $P(A-B)$ with $P(B^c)$ or to think $P(A-B) = P(A) - P(B)$. Remember: $A-B$ is only the part of $A$ that excludes $B$, not the whole complement of $B$.

   Tip

   Visualize a Venn diagram: $A$ is a circle split into the $B$ overlap and the rest. $P(A-B)$ is the “crescent” of $A$ outside $B$. Subtracting that from $P(A)$ gives the overlap directly.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

6. COMEDK 2024Set 2024-A1 markMCQ
   Q.P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}$. The probability that $\mathrm{P}$ applies for the job given that $\mathrm{Q}$ applies for the job is $\frac{1}{2}$, and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}$. Then the probability that $\mathrm{P}$ does not apply for the job given that $\mathrm{Q}$ does not apply for the job is (A) $\frac{4}{5}$ (B) $\frac{7}{8}$ (C) $\frac{5}{6}$ (D) $\frac{11}{12}$

   ›Reveal solutionSolution

   We are given conditional probabilities and need to find $P(\overline{P} \mid \overline{Q})$. Using the definitions of conditional probability and the law of total probability, we compute $P(Q)$ and $P(\overline{Q})$, then apply Bayes' theorem to get $\frac{4}{5}$, which corresponds to option (A).

   Concept and intuition:

   This is a classic problem of working backwards from conditional probabilities to find a joint probability table. We know $P(P)$, $P(P \mid Q)$, and $P(Q \mid P)$. From these, we can find $P(P \cap Q)$ in two ways, which lets us solve for $P(Q)$. Then we can compute the desired conditional probability $P(\overline{P} \mid \overline{Q})$ using the complement rule and the definition of conditional probability.

   Step-by-step solution:

   1. Write down what is given. Let $P$ = event that P applies, $Q$ = event that Q applies. We have:

   $P(P) = \frac{1}{4}, \quad P(P \mid Q) = \frac{1}{2}, \quad P(Q \mid P) = \frac{1}{3}.$

   2. Use the definition of conditional probability to express $P(P \cap Q)$ in two ways. From $P(P \mid Q) = \frac{P(P \cap Q)}{P(Q)}$, we get

   $P(P \cap Q) = P(P \mid Q) \cdot P(Q) = \frac{1}{2} P(Q).$

   From $P(Q \mid P) = \frac{P(P \cap Q)}{P(P)}$, we get

   $P(P \cap Q) = P(Q \mid P) \cdot P(P) = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}.$

   3. Equate the two expressions for $P(P \cap Q)$ to find $P(Q)$.

   $\frac{1}{2} P(Q) = \frac{1}{12} \quad \Rightarrow \quad P(Q) = \frac{1}{6}.$

   4. Find $P(\overline{Q})$ and $P(P \cap \overline{Q})$.

   $P(\overline{Q}) = 1 - P(Q) = 1 - \frac{1}{6} = \frac{5}{6}.$

   Also, $P(P \cap \overline{Q}) = P(P) - P(P \cap Q) = \frac{1}{4} - \frac{1}{12} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6}.$

   5. Find $P(\overline{P} \cap \overline{Q})$. Since $\overline{P} \cap \overline{Q}$ is the complement of $P \cup Q$, we can use:

   $P(\overline{P} \cap \overline{Q}) = 1 - P(P \cup Q).$

   First, $P(P \cup Q) = P(P) + P(Q) - P(P \cap Q) = \frac{1}{4} + \frac{1}{6} - \frac{1}{12}.$

   Common denominator 12: $\frac{3}{12} + \frac{2}{12} - \frac{1}{12} = \frac{4}{12} = \frac{1}{3}.$

   So $P(\overline{P} \cap \overline{Q}) = 1 - \frac{1}{3} = \frac{2}{3}.$

   6. Compute the desired conditional probability.

   $P(\overline{P} \mid \overline{Q}) = \frac{P(\overline{P} \cap \overline{Q})}{P(\overline{Q})} = \frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \cdot \frac{6}{5} = \frac{12}{15} = \frac{4}{5}.$

   Watch out

   A common mistake is to assume $P(P \mid Q)$ and $P(Q \mid P)$ are reciprocals or that $P(P \cap Q)$ can be found by multiplying $P(P)$ and $P(Q)$ directly — that only works for independent events, which is not the case here.

   Tip

   You can also solve this by constructing a 2×2 probability table. From $P(P)=\frac14$ and $P(P \cap Q)=\frac1{12}$, fill in the rest systematically. The final answer is the same.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

7. KCET 2025Set A-11 markMCQ
   Q.Meera visits only one of the two temples A and B in her locality. Probability that she visits temple A is $\dfrac{2}{5}$. If she visits temple A, $\dfrac{1}{3}$ is the probability that she meets her friend, whereas it is $\dfrac{2}{7}$ if she visits temple B. Meera met her friend at one of the two temples. The probability that she met her at temple B is (A) $\dfrac{7}{16}$ (B) $\dfrac{5}{16}$ (C) $\dfrac{3}{16}$ (D) $\dfrac{9}{16}$

   ›Reveal solutionSolution

   The friend has already been met (the effect); we want the probability of the cause (temple B) — that reversal of conditioning is exactly Bayes' theorem.

   Step 1 — Name the events.

   Let $A$ = "Meera visits temple A", $B$ = "Meera visits temple B", $F$ = "she meets her friend".

   She visits only one of the two temples, so $A$ and $B$ are mutually exclusive and exhaustive:

   $P(A)=\frac25\quad\Longrightarrow\quad P(B)=1-\frac25=\frac35 .$

   Given: $P(F\mid A)=\dfrac13$, $P(F\mid B)=\dfrac27$.

   Step 2 — Why Bayes.

   We are told the outcome ($F$ happened) and asked for the probability of a cause ($B$). Bayes' theorem inverts the conditioning:

   $P(B\mid F)=\frac{P(B)\,P(F\mid B)}{P(A)\,P(F\mid A)+P(B)\,P(F\mid B)} .$

   The denominator is $P(F)$ by the law of total probability — the friend can be met on either branch.

   Step 3 — Compute the two branch probabilities.

   $P(A\cap F)=\frac25\times\frac13=\frac{2}{15},\qquad P(B\cap F)=\frac35\times\frac27=\frac{6}{35} .$

   Step 4 — Total probability of meeting the friend.

   LCM of 15 and 35 is 105:

   $\frac{2}{15}=\frac{14}{105},\qquad \frac{6}{35}=\frac{18}{105},$

   $P(F)=\frac{14}{105}+\frac{18}{105}=\frac{32}{105}.$

   Step 5 — Apply Bayes.

   $P(B\mid F)=\frac{18/105}{32/105}=\frac{18}{32}=\frac{9}{16}.$

   (Check: $P(A\mid F)=\frac{14}{32}=\frac{7}{16}$, and $\frac{9}{16}+\frac{7}{16}=1$ ✓. Note option (A) $\frac{7}{16}$ is the trap — it is the probability for temple A.)

   ✓Final answer

   The correct option is (D) — $\dfrac{9}{16}$.

   ANSWER: D

8. KCET 2021Set A-11 markMCQ
   Q.Given that A and B are two events such that $P(B) = \dfrac{3}{5}$, $P(A/B) = \dfrac{1}{2}$ and $P(A \cup B) = \dfrac{4}{5}$, then $P(A) =$ (A) $\dfrac{3}{10}$ (B) $\dfrac{1}{2}$ (C) $\dfrac{1}{5}$ (D) $\dfrac{3}{5}$

   ›Reveal solutionSolution

   Turn the conditional probability into $P(A\cap B)$, then substitute into the addition theorem and solve for $P(A)$.

   Step 1 — Extract $P(A\cap B)$ from the conditional probability.

   By definition,

   $P(A/B)=\frac{P(A\cap B)}{P(B)}\qquad (P(B)\neq0).$

   Rearranging (the multiplication theorem):

   $P(A\cap B)=P(A/B)\cdot P(B)=\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}.$

   This is the key move — the conditional probability is not directly usable in the union formula, but $P(A\cap B)$ is.

   Step 2 — Apply the addition theorem.

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

   The intersection is subtracted because the elements common to $A$ and $B$ would otherwise be counted twice.

   Step 3 — Substitute the known values.

   $\frac{4}{5}=P(A)+\frac{3}{5}-\frac{3}{10}.$

   Step 4 — Solve for $P(A)$.

   $P(A)=\frac{4}{5}-\frac{3}{5}+\frac{3}{10}=\frac{1}{5}+\frac{3}{10}=\frac{2}{10}+\frac{3}{10}=\frac{5}{10}=\frac{1}{2}.$

   Consistency check. With $P(A)=\tfrac12$, $P(B)=\tfrac35$, $P(A\cap B)=\tfrac{3}{10}$: note $P(A)P(B)=\tfrac12\cdot\tfrac35=\tfrac{3}{10}=P(A\cap B)$, so $A$ and $B$ are independent — entirely consistent with $P(A/B)=\tfrac12=P(A)$. And $P(A\cup B)=\tfrac12+\tfrac35-\tfrac3{10}=\tfrac{5+6-3}{10}=\tfrac{8}{10}=\tfrac45$. ✓

   ✓Final answer

   The correct option is (B) — $\dfrac{1}{2}$.

   ANSWER: B

9. KCET 2025Set A-11 markMCQ
   Q.If A and B are two non-mutually exclusive events such that $P(A|B) = P(B|A)$, then (A) $A \subset B$ but $A \neq B$ (B) $A = B$ (C) $A \cap B = \phi$ (D) $P(A) = P(B)$

   ›Reveal solutionSolution

   Write both conditional probabilities with the same numerator $P(A \cap B)$, cancel it (legal because the events are not mutually exclusive), and the denominators must be equal.

   Step 1 — Definition of conditional probability.

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

   Note $A \cap B = B \cap A$, so the two fractions share the same numerator.

   Step 2 — Impose the given condition.

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

   Step 3 — Cancel — and see why we are allowed to.

   The events are non-mutually-exclusive, i.e. $A \cap B \ne \phi$ and $P(A \cap B) \ne 0$. (This hypothesis is exactly what makes the cancellation valid — if $P(A\cap B)$ were 0, both sides would be 0 for any $P(A), P(B)$ and nothing would follow.) Dividing both sides by $P(A\cap B)$:

   $\frac{1}{P(B)} = \frac{1}{P(A)} \;\Longrightarrow\; P(A) = P(B).$

   Step 4 — Why the other options are not forced.

   Equality of probabilities does not force equality of sets. Example: toss a fair coin twice; let $A$ = "first toss is head", $B$ = "second toss is head". Then $P(A) = P(B) = 1/2$, $P(A\cap B) = 1/4 \neq 0$, and indeed $P(A|B) = P(B|A) = 1/2$ — yet $A \ne B$ and neither is a subset of the other. So (A) and (B) are not implied. (C) contradicts the given non-mutual-exclusivity.

   ✓Final answer

   The correct option is (D) — $P(A) = P(B)$.

   ANSWER: D

10. COMEDK 2024Set 2024-E1 markMCQ
    Q.Let $\mathrm{A}$ and ${B}$ be two events such that $P(A / B)=\frac{1}{2}$ and $P(B / A)=\frac{1}{3}$ and $P(A \cap B)=\frac{1}{6}$ then, which one of the following is not true? (A) $\mathrm{A} \text { and } \mathrm{B} \text { are not independent }$ (B) $P(A \cup B)=\frac{2}{3}$ (C) $P\left(A^{\prime} \cap B\right)=\frac{1}{6}$ (D) $A \text { and } B \text { are independent }$

    ›Reveal solutionSolution

    $P(A)=\tfrac12,\ P(B)=\tfrac13$, and $P(A)P(B)=\tfrac16=P(A\cap B)$, so $A,B$ ARE independent; the false statement is that they are not independent.

    From $P(A\cap B)=P(A\mid B)\,P(B)$:

    $\frac{1}{6}=\frac{1}{2}\,P(B)\implies P(B)=\frac{1}{3}$

    From $P(A\cap B)=P(B\mid A)\,P(A)$:

    $\frac{1}{6}=\frac{1}{3}\,P(A)\implies P(A)=\frac{1}{2}$

    Independence test:

    $P(A)\,P(B)=\frac{1}{2}\cdot\frac{1}{3}=\frac{1}{6}=P(A\cap B)$

    So $A$ and $B$ are independent. Checking the other options:

    $P(A\cup B)=\frac12+\frac13-\frac16=\frac{2}{3}\quad(\text{B true})$

    $P(A'\cap B)=P(B)-P(A\cap B)=\frac13-\frac16=\frac{1}{6}\quad(\text{C true})$

    Option (D) "$A$ and $B$ are independent" is true. Therefore the statement that is not true is (A) "$A$ and $B$ are not independent".

    ✓Final answer

    The statement that is not true is "$A$ and $B$ are not independent" — option (A).

11. COMEDK 2026Set 2026-M1 markMCQ
    Q.Advika chooses one of three scarves every morning: Red, Blue, or Green. The probability she chooses Red is $20 \%$. The probability she chooses Blue is twice the probability of choosing Red. On the remaining days she wears a Green scarf. Once a scarf is chosen, she decides whether to wear a Hat (H) and Sunglasses (S). These choices are independent of each other but depend on the scarf colour: $\begin{array}{|l|l|l|} \hline \text { Scarf colour } & \mathbf{P ( H )} & \mathbf{P ( S )} \\ \hline \text { Red } & 0.5 & 0.8 \\ \hline \text { Blue } & 0.4 & 0.5 \\ \hline \text { Green } & 0.1 & 0.5 \\ \hline \end{array}$ Advika is spotted outdoors wearing both a Hat and Sunglasses. What is the probability that she is wearing the Red scarf? (A) $\frac{13}{313}$ (B) $\frac{8}{21}$ (C) $\frac{4}{9}$ (D) $\frac{8}{13}$

    ›Reveal solutionSolution

    Bayes' theorem on scarf colour given that both a hat and sunglasses are worn. Priors $P(R)=0.2,\ P(B)=0.4,\ P(G)=0.4$; likelihoods $P(H\cap S\mid\text{colour})=P(H)\,P(S)$. The posterior $P(R\mid H\cap S)=\frac{4}{9}$ — option (C).

    Concept. Hat and sunglasses are independent given the scarf, so $P(H\cap S\mid\text{colour})=P(H\mid\text{colour})\cdot P(S\mid\text{colour})$. Bayes' theorem then reverses the conditioning to give the probability of the scarf colour from the observed accessories.

    Step 1 — Priors.

    $P(R)=20\%=0.2,\qquad P(B)=2P(R)=0.4,\qquad P(G)=1-0.2-0.4=0.4.$

    Step 2 — Likelihood of wearing both accessories for each colour.

    $P(H\cap S\mid R)=0.5\times0.8=0.40,$

    $P(H\cap S\mid B)=0.4\times0.5=0.20,$

    $P(H\cap S\mid G)=0.1\times0.5=0.05.$

    Step 3 — Total probability of both accessories (denominator).

    $P(H\cap S)=(0.2)(0.40)+(0.4)(0.20)+(0.4)(0.05)=0.08+0.08+0.02=0.18.$

    Step 4 — Posterior for Red.

    $P(R\mid H\cap S)=\frac{P(R)\,P(H\cap S\mid R)}{P(H\cap S)}=\frac{0.08}{0.18}=\frac{8}{18}=\frac{4}{9}.$

    ✓Final answer

    $P(\text{Red}\mid H\cap S)=\dfrac{4}{9}$ — option (C).

    ANSWER: C

12. COMEDK 2023Set 2023-E1 markMCQ
    Q.

    [!FORMULA] $\text { If } P(B)=\frac{3}{5} \quad P(A / B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5} \text { then } P(A \cup B)^{\prime}+P\left(A^{\prime} \cup B\right)=$

    (A) $\frac{4}{5}$ (B) $\frac{1}{2}$ (C) 1 (D) $\frac{1}{5}$

    ›Reveal solutionSolution

    Step 5: sum = 1/5 + 4/5 = 1.

    Concept: conditional probability, addition theorem, complements, De Morgan.

    Given P(B) = 3/5, P(A|B) = 1/2, P(A U B) = 4/5.

    Step 1: P(A ^ B) = P(A|B) P(B) = (1/2)(3/5) = 3/10.

    Step 2: from P(A U B) = P(A) + P(B) - P(A ^ B),

    P(A) = 4/5 - 3/5 + 3/10 = 1/5 + 3/10 = 1/2.

    Step 3: P((A U B)') = 1 - P(A U B) = 1 - 4/5 = 1/5.

    Step 4: P(A' U B) = 1 - P((A' U B)') = 1 - P(A ^ B') [De Morgan]

    P(A ^ B') = P(A) - P(A ^ B) = 1/2 - 3/10 = 1/5.

    So P(A' U B) = 1 - 1/5 = 4/5.

    Step 5: sum = 1/5 + 4/5 = 1.

    ✓Final answer

    The correct option is (C) — 1

    ANSWER: C