Q.Three events $A$, $B$ and $C$ have probabilities $\dfrac{2}{5}$, $\dfrac{1}{3}$ and $\dfrac{1}{2}$, respectively. Given that $P(A \cap C) = \dfrac{1}{5}$ and $P(B \cap C) = \dfrac{1}{4}$, find the values of $P(C \mid B)$ and $P(A' \cap C')$.

Showing the 12 most recent of 30 on this concept.

1. TG EAPCET 2026Set eng-2026-05-09-FN1 markMCQ
   Q.If $P(A) = \dfrac{3}{8}$, $P(\overline{A}|B) = P(\overline{B}|A) = \dfrac{3}{5}$, then $P(A \cap B) + P(B) =$ (A) $\dfrac{21}{40}$ (B) $\dfrac{2}{13}$ (C) $\dfrac{3}{14}$ (D) $\dfrac{5}{12}$

   ›Reveal solutionSolution

   We use the given conditional probabilities to set up equations for $P(A \cap B)$ and $P(B)$, then solve and sum them. The result is $\frac{21}{40}$, which corresponds to option (A).

   We are told:

   • $P(A) = \frac{3}{8}$
   • $P(\overline{A} \mid B) = \frac{3}{5}$
   • $P(\overline{B} \mid A) = \frac{3}{5}$

   We need $P(A \cap B) + P(B)$.

   ---

   Concept and intuition

   Conditional probabilities like $P(\overline{A} \mid B)$ relate the probability of the complement of $A$ given $B$ to the joint probability $P(\overline{A} \cap B)$. Since $P(\overline{A} \mid B) = \frac{P(\overline{A} \cap B)}{P(B)}$, we can write an equation linking $P(B)$ and $P(A \cap B)$. Similarly, $P(\overline{B} \mid A)$ gives a relation between $P(A)$ and $P(A \cap B)$. This lets us solve for the unknowns.

   ---

   Step-by-step solution

   1. Use $P(\overline{B} \mid A)$ to find $P(A \cap B)$ By definition:

   $P(\overline{B} \mid A) = \frac{P(\overline{B} \cap A)}{P(A)} = \frac{3}{5}$

   Since $P(\overline{B} \cap A) = P(A) - P(A \cap B)$, we have:

   $\frac{P(A) - P(A \cap B)}{P(A)} = \frac{3}{5}$

   Substitute $P(A) = \frac{3}{8}$:

   $\frac{\frac{3}{8} - P(A \cap B)}{\frac{3}{8}} = \frac{3}{5}$

   Multiply both sides by $\frac{3}{8}$:

   $\frac{3}{8} - P(A \cap B) = \frac{3}{5} \cdot \frac{3}{8} = \frac{9}{40}$

   So:

   $P(A \cap B) = \frac{3}{8} - \frac{9}{40} = \frac{15}{40} - \frac{9}{40} = \frac{6}{40} = \frac{3}{20}$

   2. Use $P(\overline{A} \mid B)$ to find $P(B)$ By definition:

   $P(\overline{A} \mid B) = \frac{P(\overline{A} \cap B)}{P(B)} = \frac{3}{5}$

   Now $P(\overline{A} \cap B) = P(B) - P(A \cap B)$. Substitute $P(A \cap B) = \frac{3}{20}$:

   $\frac{P(B) - \frac{3}{20}}{P(B)} = \frac{3}{5}$

   Multiply both sides by $P(B)$: …

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2. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
   Q.A, $B_1$, $B_2$, $B_3$ are the events in a random experiment. If $P(B_1) = 0.25$, $P(B_2) = 0.30$, $P(B_3) = 0.45$, $P\left(\dfrac{A}{B_1}\right) = 0.05$, $P\left(\dfrac{A}{B_2}\right) = 0.04$, $P\left(\dfrac{A}{B_3}\right) = 0.03$, then $P\left(\dfrac{B_2}{A}\right) =$ (A) $\dfrac{6}{19}$ (B) $\dfrac{8}{19}$ (C) $\dfrac{12}{19}$ (D) $\dfrac{5}{19}$

   ›Reveal solutionSolution

   This is a classic Bayes’ theorem problem: we are given prior probabilities for three mutually exclusive events and the conditional probabilities of A given each, and we need the posterior probability of $B_2$ given A. The answer is $\frac{6}{19}$, which corresponds to option (A).

   We start with the concept: Bayes’ theorem lets us “reverse” conditional probabilities. Here, we know $P(A \mid B_i)$ and want $P(B_2 \mid A)$. The key is that the $B_i$ form a partition of the sample space (they are the only possible “causes” of A), so we can compute $P(A)$ using the law of total probability, then apply Bayes’ formula.

   1. Identify the given data

      • $P(B_1) = 0.25$, $P(B_2) = 0.30$, $P(B_3) = 0.45$
      • $P(A \mid B_1) = 0.05$, $P(A \mid B_2) = 0.04$, $P(A \mid B_3) = 0.03$ The events $B_1, B_2, B_3$ are mutually exclusive and exhaustive (their probabilities sum to 1), so they form a partition.

   2. Compute the total probability of A

      By the law of total probability:

   $P(A) = P(B_1)P(A \mid B_1) + P(B_2)P(A \mid B_2) + P(B_3)P(A \mid B_3)$

   Substitute:

   $P(A) = (0.25)(0.05) + (0.30)(0.04) + (0.45)(0.03)$

   Calculate each term:

   • $0.25 \times 0.05 = 0.0125$
   • $0.30 \times 0.04 = 0.0120$
   • $0.45 \times 0.03 = 0.0135$ Sum:

   $P(A) = 0.0125 + 0.0120 + 0.0135 = 0.0380$

   3. Apply Bayes’ theorem for $P(B_2 \mid A)$ Bayes’ theorem states: $$$$

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3. TG EAPCET 2026Set eng-2026-05-10-FN1 markMCQ
   Q.Three persons A, B, C planned to have a running race among themselves. If the probability that A wins the race is thrice that of B and the probability that B wins the race is $\dfrac{3}{2}$ times that of C, then the difference in probabilities of A and C to win the race is (A) $\dfrac{2}{3}$ (B) $\dfrac{1}{2}$ (C) $\dfrac{5}{14}$ (D) $\dfrac{3}{7}$

   ›Reveal solutionSolution

   With $P(A)=\tfrac{9}{14},\ P(C)=\tfrac{2}{14}$, the difference is $P(A)-P(C)=\dfrac12$.

   Let $P(C)=p$. Then $P(B)=\dfrac{3}{2}p$ and $P(A)=3\,P(B)=\dfrac{9}{2}p$.

   The three probabilities sum to $1$ (one of them must win):

   $\frac{9}{2}p+\frac{3}{2}p+p=7p=1\ \Rightarrow\ p=\frac{1}{7}.$

   Hence

   $P(A)=\frac{9}{2}\cdot\frac17=\frac{9}{14},\qquad P(C)=\frac17=\frac{2}{14}.$ …

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4. TG EAPCET 2022Set eng-2022-07-19-FN1 markMCQ
   Q.If $A$ and $B$ are two events of a random experiment such that $P(\overline{A}) = \dfrac{2}{3}$, $P(B) = \dfrac{4}{15}$ and $P(A \cap \overline{B}) = \dfrac{1}{5}$, then $\sqrt{195 \left[ P\left(B \mid (A \cup \overline{B})\right) + P(A \cup B) \right]} =$ (A) 9 (B) 11 (C) 13 (D) 15

   ›Reveal solutionSolution

   This problem requires us to calculate probabilities of various event combinations (complement, intersection, union) and a conditional probability using fundamental set theory identities. We then substitute these values into the given expression to find the final numerical result. The final value is $\boxed{11}$.

   To solve this problem, we need to systematically break down the given expression and calculate each probability term using the fundamental rules of probability and set theory. The key is to correctly apply the formulas for complements, unions, intersections, and conditional probabilities, often using set identities to simplify complex event descriptions.

   Here's a step-by-step approach:

   1. Determine $P(A)$ from $P(\overline{A})$:

      The probability of an event $A$ and its complement $\overline{A}$ always sum to 1.

      $P(A) + P(\overline{A}) = 1$

      Given $P(\overline{A}) = \frac{2}{3}$, we can find $P(A)$:

      $P(A) = 1 - P(\overline{A}) = 1 - \frac{2}{3} = \frac{1}{3}$.

   2. Determine $P(A \cap B)$ using $P(A \cap \overline{B})$:

      The event $A$ can be partitioned into two mutually exclusive events: $A \cap B$ (A and B both occur) and $A \cap \overline{B}$ (A occurs, but B does not).

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

      We are given $P(A \cap \overline{B}) = \frac{1}{5}$ and we found $P(A) = \frac{1}{3}$.

      So, $P(A \cap B) = P(A) - P(A \cap \overline{B}) = \frac{1}{3} - \frac{1}{5}$.

      To subtract these fractions, we find a common denominator, which is 15:

      $P(A \cap B) = \frac{5}{15} - \frac{3}{15} = \frac{2}{15}$.

   3. Calculate $P(A \cup B)$:

      The probability of the union of two events $A$ and $B$ is given by the addition rule.

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

      We have $P(A) = \frac{1}{3}$, $P(B) = \frac{4}{15}$ (given), and $P(A \cap B) = \frac{2}{15}$ (from Step 2).

      $P(A \cup B) = \frac{1}{3} + \frac{4}{15} - \frac{2}{15}$.

      Using a common denominator of 15:

      $P(A \cup B) = \frac{5}{15} + \frac{4}{15} - \frac{2}{15} = \frac{5+4-2}{15} = \frac{7}{15}$.

      This is the first part of the sum inside the square root.

   4. Prepare for $P(B \mid (A \cup \overline{B}))$: Identify the intersection term:

      The conditional probability $P(X \mid Y)$ is defined as $\frac{P(X \cap Y)}{P(Y)}$. Here, $X=B$ and $Y=(A \cup \overline{B})$.

      So, we need to find $P(B \cap (A \cup \overline{B}))$.

      Using the distributive property of set intersection over union:

      $B \cap (A \cup \overline{B}) = (B \cap A) \cup (B \cap \overline{B})$.

      The event $B \cap \overline{B}$ means that event $B$ occurs AND event $B$ does NOT occur, which is impossible. Thus, $B \cap \overline{B} = \emptyset$.

      So, $B \cap (A \cup \overline{B}) = (B \cap A) \cup \emptyset = B \cap A$.

      Therefore, $P(B \cap (A \cup \overline{B})) = P(A \cap B)$.

      From Step 2, we know $P(A \cap B) = \frac{2}{15}$. This is the numerator for our conditional probability.

   5. Prepare for $P(B \mid (A \cup \overline{B}))$: Calculate $P(\overline{B})$:

      Similar to Step 1, we use the complement rule for event $B$.

      $P(\overline{B}) = 1 - P(B)$.

      Given $P(B) = \frac{4}{15}$:

      $P(\overline{B}) = 1 - \frac{4}{15} = \frac{15-4}{15} = \frac{11}{15}$.

   6. Prepare for $P(B \mid (A \cup \overline{B}))$: Calculate $P(A \cup \overline{B})$:

      This is the denominator for our conditional probability. We use the addition rule for $A$ and $\overline{B}$.

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

      We have $P(A) = \frac{1}{3}$ (from Step 1), $P(\overline{B}) = \frac{11}{15}$ (from Step 5), and $P(A \cap \overline{B}) = \frac{1}{5}$ (given).

      $P(A \cup \overline{B}) = \frac{1}{3} + \frac{11}{15} - \frac{1}{5}$.

      Using a common denominator of 15: …

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5. TG EAPCET 2021Set eng-2021-08-04-FN1 markMCQ
   Q.If $P\left(\dfrac{A}{B}\right) = \dfrac{3}{10}$, $P\left(\dfrac{B}{A}\right) = \dfrac{4}{5}$ and $P(A \cup B) = K P(B)$, then $\dfrac{1}{K} =$ (A) $\dfrac{40}{49}$ (B) $\dfrac{40}{43}$ (C) $\dfrac{100}{101}$ (D) $1$

   ›Reveal solutionSolution

   The key idea is to use the definitions of conditional probability to relate $P(A \cap B)$ to $P(A)$ and $P(B)$, then express $P(A \cup B)$ in terms of $P(B)$ alone. The result is $\frac{1}{K} = \frac{40}{43}$.

   We are given two conditional probabilities and a relation involving the union. The goal is to find $\frac{1}{K}$, where $P(A \cup B) = K P(B)$. This is a problem about linking conditional probabilities to the basic probability of events, so we start by writing down what each conditional means.

   Recall: $P(A/B) = \frac{P(A \cap B)}{P(B)}$ and $P(B/A) = \frac{P(A \cap B)}{P(A)}$. These are not symmetric — each gives a different ratio. Our job is to use them to find $P(A)$ and $P(A \cap B)$ in terms of $P(B)$, then compute $P(A \cup B)$.

   1. From $P(A/B)$ we get $P(A \cap B)$ in terms of $P(B)$.

   $P(A/B) = \frac{3}{10} \quad\Rightarrow\quad \frac{P(A \cap B)}{P(B)} = \frac{3}{10}$

   So

   $P(A \cap B) = \frac{3}{10} P(B).$

   2. From $P(B/A)$ we get $P(A)$ in terms of $P(A \cap B)$.

   $P(B/A) = \frac{4}{5} \quad\Rightarrow\quad \frac{P(A \cap B)}{P(A)} = \frac{4}{5}$

   Hence

   $P(A) = \frac{5}{4} P(A \cap B).$

   3. Substitute the expression for $P(A \cap B)$ from step 1 into step 2.

   $P(A) = \frac{5}{4} \cdot \frac{3}{10} P(B) = \frac{15}{40} P(B) = \frac{3}{8} P(B).$

   So $P(A)$ is $\frac{3}{8}$ of $P(B)$.

   4. Now write $P(A \cup B)$ using the inclusion-exclusion formula.

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

   Substitute the expressions we have:

   $P(A \cup B) = \frac{3}{8} P(B) + P(B) - \frac{3}{10} P(B).$

   5. Combine the terms over a common denominator. …

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6. TG EAPCET 2026Set eng-2026-05-10-AN1 markMCQ
   Q.A card is drawn randomly from a well shuffled pack of 52 cards. If A is the event of getting a diamond card and B is the event of getting an ace card, then the probability that exactly one of the events among A and B to occur is (A) $\dfrac{15}{52}$ (B) $\dfrac{4}{13}$ (C) $\dfrac{17}{52}$ (D) $\dfrac{5}{13}$

   ›Reveal solutionSolution

   The probability that exactly one of the events A (diamond) or B (ace) occurs is the sum of their individual probabilities minus twice the probability of both occurring. The result is $\frac{15}{52}$, which corresponds to option (A).

   We want the probability that exactly one of the two events happens — that is, either we draw a diamond that is not an ace, or we draw an ace that is not a diamond. This is a classic "exclusive or" (XOR) situation.

   Why this approach works:

   If we simply add $P(A) + P(B)$, we count the case where both occur (the ace of diamonds) twice. To get exactly one, we subtract that double-counted overlap once more than usual — hence $P(A) + P(B) - 2P(A \cap B)$.

   ---

   1. Identify the probabilities of each event individually.

      • There are 13 diamonds in a deck of 52, so $P(A) = \frac{13}{52} = \frac{1}{4}$.
      • There are 4 aces, so $P(B) = \frac{4}{52} = \frac{1}{13}$.

   2. Find the probability that both events occur (the intersection).

      • Only one card is both a diamond and an ace: the ace of diamonds.
      • So $P(A \cap B) = \frac{1}{52}$.

   3. Apply the formula for exactly one event.

      • Exactly one of A or B occurs means: $(A \text{ and not } B)$ or $(B \text{ and not } A)$.
      • The probability is:

   $P(\text{exactly one}) = P(A) + P(B) - 2P(A \cap B)$

   • Substitute the values:

   $\frac{13}{52} + \frac{4}{52} - 2 \cdot \frac{1}{52} = \frac{13 + 4 - 2}{52} = \frac{15}{52}$

   4. Check against the options. …

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7. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
   Q.Let $P = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ be a matrix. Three elements of this matrix P are selected at random. A is the event of having the three elements whose sum is odd. B is the event of selecting the three elements which are in a row or column. Then $P(A) + P\left(\dfrac{A}{B}\right) =$ (A) $\dfrac{221}{420}$ (B) $\dfrac{17}{21}$ (C) $\dfrac{21}{20}$ (D) $\dfrac{3}{2}$

   ›Reveal solutionSolution

   We compute the probability that three randomly chosen entries sum to an odd number, then the conditional probability of that event given they lie in a single row or column, and add them. The result simplifies to $\frac{17}{21}$, which is option (B).

   ---

   Concept & Intuition

   The matrix has 9 entries. We pick 3 of them uniformly at random.

   • For event A (odd sum), we need to count how many triples have an odd total. Since odd/even depends only on parity, we first classify the 9 numbers by parity: 1,3,5,7,9 are odd (5 odds); 2,4,6,8 are even (4 evens). The sum of three numbers is odd iff we have an odd number of odd entries among them — i.e., 1 or 3 odds.
   • For event B (all in one row or one column), we count triples that lie entirely in a single row (3 rows, each row has 3 entries → 1 triple per row) or entirely in a single column (3 columns, each column has 3 entries → 1 triple per column). That gives $3+3=6$ triples.
   • Then $P(A/B)$ is the fraction of those 6 triples that also have an odd sum.

   We compute both probabilities and add them.

   ---

   Step-by-step

   1. Total number of ways to choose 3 entries from 9

   $\binom{9}{3} = 84.$

   2. Count triples with odd sum (event A)
      • Case 1: exactly 1 odd, 2 evens Choose 1 odd from 5 odds: $\binom{5}{1}=5$ Choose 2 evens from 4 evens: $\binom{4}{2}=6$ Total: $5 \times 6 = 30$ triples.
      • Case 2: exactly 3 odds, 0 evens Choose 3 odds from 5 odds: $\binom{5}{3}=10$ Total: $10$ triples.
      • So $|A| = 30 + 10 = 40$. Hence

   $P(A) = \frac{40}{84} = \frac{10}{21}.$

   3. Count triples in a row or column (event B)
      • Rows: 3 rows, each row has exactly 1 triple (all three entries). So 3 triples.
      • Columns: 3 columns, each column has exactly 1 triple. So 3 triples.
      • Total: $|B| = 6$. Hence

   $P(B) = \frac{6}{84} = \frac{1}{14}.$

   4. Count triples that are both in a row/column AND have odd sum (event $A \cap B$) Check each row and column for odd sum: …

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8. TG EAPCET 2022Set eng-2022-07-20-AN1 markMCQ
   Q.If $X$ is a Poisson variate such that $\frac{5}{3}k = P(X=2) = P(X=3)$, then $P(X=5)=$ (A) $k$ (B) $\frac{1}{4}k$ (C) $\frac{1}{2}k$ (D) $\frac{3}{4}k$

   ›Reveal solutionSolution

   We use the Poisson Probability Mass Function to equate $P(X=2)$ and $P(X=3)$, which allows us to determine the parameter $\lambda$. Once $\lambda$ is known, we can express $P(X=5)$ in terms of $k$. The result is $P(X=5) = \boxed{\frac{3}{4}k}$.

   The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence and independence of events. It is characterized by a single parameter, $\lambda$ (lambda), which represents the average number of events in the given interval.

   The core idea here is to use the given equality of probabilities, $P(X=2) = P(X=3)$, to find the value of this parameter $\lambda$. Once $\lambda$ is known, we can calculate any other probability $P(X=x)$ using the Poisson Probability Mass Function (PMF). We are also given a relationship involving $k$, which we will use to express our final answer in terms of $k$.

   1. Recall the Poisson Probability Mass Function (PMF):

      For a Poisson variate $X$ with parameter $\lambda$, the probability of observing exactly $x$ events is given by:

      $P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!}$

      where $x = 0, 1, 2, \dots$ and $\lambda > 0$.

   2. Use the given condition $P(X=2) = P(X=3)$ to find $\lambda$:

      Substitute $x=2$ and $x=3$ into the PMF:

      $P(X=2) = \frac{e^{-\lambda} \lambda^2}{2!}$

      $P(X=3) = \frac{e^{-\lambda} \lambda^3}{3!}$

      Equating these two probabilities:

      $\frac{e^{-\lambda} \lambda^2}{2!} = \frac{e^{-\lambda} \lambda^3}{3!}$

      Since $e^{-\lambda}$ is never zero and $\lambda$ must be positive (as probabilities are non-zero), we can divide both sides by $e^{-\lambda} \lambda^2$:

      $\frac{1}{2!} = \frac{\lambda}{3!}$

      Recall that $2! = 2 \times 1 = 2$ and $3! = 3 \times 2 \times 1 = 6$.

      $\frac{1}{2} = \frac{\lambda}{6}$

      Multiply both sides by 6 to solve for $\lambda$:

      $\lambda = \frac{6}{2} = 3$

      So, the parameter of the Poisson distribution is $\lambda = 3$.

   3. Express $k$ in terms of $\lambda$ (and $e^{-\lambda}$):

      We are given that $\frac{5}{3}k = P(X=2)$.

      We know $P(X=2) = \frac{e^{-\lambda} \lambda^2}{2!}$. Substitute $\lambda=3$:

      $P(X=2) = \frac{e^{-3} 3^2}{2} = \frac{9e^{-3}}{2}$

      Now, equate this to $\frac{5}{3}k$:

      $\frac{5}{3}k = \frac{9e^{-3}}{2}$

      Solve for $k$:

      $k = \frac{3}{5} \times \frac{9e^{-3}}{2} = \frac{27e^{-3}}{10}$

   4. Calculate $P(X=5)$ using $\lambda=3$: …

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9. TG EAPCET 2026Set eng-2026-05-09-AN1 markMCQ
   Q.In a Poisson distribution with parameter $\lambda$, if $5P(X=3) = P(X=5)$, then $P(X=2) =$ (A) $\dfrac{25}{e^5}$ (B) $\dfrac{50}{e^{10}}$ (C) $\dfrac{30}{e^6}$ (D) $\dfrac{40}{e^8}$

   ›Reveal solutionSolution

   The key idea is to use the Poisson probability formula $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$ and the given relation $5P(X=3) = P(X=5)$ to solve for $\lambda$, then compute $P(X=2)$. The final answer is $\frac{50}{e^{10}}$, which corresponds to option (B).

   The Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate $\lambda$. The probability mass function is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$ for $k = 0, 1, 2, \dots$. Here, the condition $5P(X=3) = P(X=5)$ gives a direct equation in $\lambda$ because the $e^{-\lambda}$ factor cancels, leaving a simple algebraic relation. Once $\lambda$ is found, plugging into $P(X=2)$ yields the answer.

   1. Write the given condition using the Poisson formula:

   $5 \cdot \frac{e^{-\lambda} \lambda^3}{3!} = \frac{e^{-\lambda} \lambda^5}{5!}$

   The factor $e^{-\lambda}$ cancels on both sides (since $\lambda$ is finite), giving:

   $5 \cdot \frac{\lambda^3}{6} = \frac{\lambda^5}{120}$

   2. Simplify the equation. Multiply both sides by 120 to clear denominators:

   $5 \cdot \frac{\lambda^3}{6} \cdot 120 = \lambda^5$

   Compute $120/6 = 20$, so $5 \cdot 20 \cdot \lambda^3 = \lambda^5$, i.e., $100 \lambda^3 = \lambda^5$.

   3. Assuming $\lambda > 0$ (since it's a rate parameter), divide both sides by $\lambda^3$:

   $100 = \lambda^2$

   Hence $\lambda = 10$ (we take the positive root because $\lambda$ is a mean, always positive). …

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10. TG EAPCET 2023Set eng-2023-05-12-FN1 markMCQ
    Q.Two numbers b and c are chosen at random in succession without replacement from the set $\{1,2,3,\ldots,9\}$. Then the probability that $x^2 + bx + c > 0,\ \forall x \in \mathbb{R}$ is (A) $\dfrac{29}{72}$ (B) $\dfrac{32}{81}$ (C) $\dfrac{45}{143}$ (D) $\dfrac{82}{125}$

    ›Reveal solutionSolution

    The condition $x^2+bx+c>0$ for all real $x$ is equivalent to the discriminant $b^2-4c<0$. Counting ordered pairs $(b,c)$ from $\{1,\dots,9\}$ without replacement that satisfy $b^2<4c$ gives $29$ favorable outcomes out of $72$ total, so the probability is $\frac{29}{72}$, which is option (A).

    ---

    Why this approach works

    A quadratic $x^2+bx+c$ that is always positive (for every real $x$) must have no real roots and open upward. Since the coefficient of $x^2$ is $1>0$, the condition reduces to the discriminant being negative: $b^2-4c<0$, i.e. $b^2<4c$.

    We are choosing $b$ and $c$ without replacement from $\{1,\dots,9\}$, so each ordered pair $(b,c)$ with $b\neq c$ is equally likely. The total number of such ordered pairs is $9\times 8 = 72$. We just need to count how many of them satisfy $b^2<4c$.

    ---

    Step-by-step counting

    1. Understand the inequality

    We need $b^2 < 4c$. Since $c$ is an integer from $1$ to $9$, rewrite as $c > \frac{b^2}{4}$. For each $b$, we count the number of $c$ values (different from $b$) that are strictly greater than $b^2/4$.

    2. Compute for each $b$

    • $b=1$: $b^2/4 = 0.25$, so $c > 0.25$ means $c \ge 1$. All $c$ from $1$ to $9$ except $c=1$ (since $b\neq c$) work. That gives $8$ choices.
    • $b=2$: $b^2/4 = 1$, so $c > 1$ means $c \ge 2$. Excluding $c=2$ leaves $\{3,4,5,6,7,8,9\}$ → $7$ choices.
    • $b=3$: $b^2/4 = 2.25$, so $c > 2.25$ means $c \ge 3$. Excluding $c=3$ leaves $\{4,5,6,7,8,9\}$ → $6$ choices.
    • $b=4$: $b^2/4 = 4$, so $c > 4$ means $c \ge 5$. Excluding $c=4$ (which isn't in this set anyway) gives $\{5,6,7,8,9\}$ → $5$ choices.
    • $b=5$: $b^2/4 = 6.25$, so $c > 6.25$ means $c \ge 7$. Excluding $c=5$ (not in set) gives $\{7,8,9\}$ → $3$ choices.
    • $b=6$: $b^2/4 = 9$, so $c > 9$ means $c \ge 10$, but max $c$ is $9$. No $c$ works → $0$ choices. …

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11. TG EAPCET 2024Set eng-2024-05-11-FN1 markMCQ
    Q.The numbers 2, 3, 5, 7, 11, 13 are written on six distinct paper chits. If 3 of them are chosen at random, then the probability that the sum of the numbers on the obtained chits is divisible by 3, is (A) $\dfrac{7}{20}$ (B) $\dfrac{6}{20}$ (C) $\dfrac{5}{20}$ (D) $\dfrac{1}{5}$

    ›Reveal solutionSolution

    The key idea is to classify each number by its remainder modulo 3, then count only those 3‑card combinations whose remainders sum to a multiple of 3. The probability is $\frac{7}{20}$, which corresponds to option (A).

    ---

    We have six numbers: 2, 3, 5, 7, 11, 13.

    We pick 3 at random. The total number of ways is $\binom{6}{3} = 20$.

    We want the probability that the sum of the three chosen numbers is divisible by 3.

    Why classify by remainder?

    A number’s remainder modulo 3 determines whether it contributes 0, 1, or 2 to the total sum mod 3. The sum of three numbers is divisible by 3 exactly when the sum of their remainders is 0 mod 3. This turns a problem about specific numbers into a simple counting problem about remainder classes.

    ---

    Step-by-step

    1. Find each number’s remainder mod 3

       • $2 \equiv 2$
       • $3 \equiv 0$
       • $5 \equiv 2$
       • $7 \equiv 1$
       • $11 \equiv 2$
       • $13 \equiv 1$

       So we have:

       • Remainder 0: {3} → 1 number
       • Remainder 1: {7, 13} → 2 numbers
       • Remainder 2: {2, 5, 11} → 3 numbers

    2. Which remainder combinations sum to 0 mod 3?

       Let (r1, r2, r3) be the remainders of the three chosen numbers. We need $r_1 + r_2 + r_3 \equiv 0 \pmod{3}$.

       The possible triples (order doesn’t matter) are:

       • (0,0,0) — all three have remainder 0
       • (1,1,1) — all three have remainder 1
       • (2,2,2) — all three have remainder 2
       • (0,1,2) — one of each remainder

       No other triple works (e.g., (0,0,1) sums to 1, etc.).

    3. Count the number of 3‑card combinations for each case

       • (0,0,0): Only 1 number with remainder 0, so impossible. Count = 0. …

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12. TG EAPCET 2025Set eng-2025-05-04-AN1 markMCQ
    Q.In a Poisson distribution, if $\dfrac{P(X=5)}{P(X=2)} = \dfrac{1}{7500}$ and $\dfrac{P(X=5)}{P(X=3)} = \dfrac{1}{500}$, then the mean of the distribution is (A) $\dfrac{1}{15}$ (B) $\dfrac{1}{5}$ (C) $\dfrac{1}{25}$ (D) $\dfrac{1}{3}$

    ›Reveal solutionSolution

    The key idea is to use the ratio formulas for Poisson probabilities to eliminate the common factor and solve for the mean λ. The mean is found to be 1/5, so option (B) is correct.

    The Poisson distribution has probability mass function

    $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0,1,2,\dots$

    where $\lambda > 0$ is the mean. When we take ratios of probabilities, the factor $e^{-\lambda}$ cancels, leaving only powers of $\lambda$ and factorials. This makes ratios a clean way to solve for $\lambda$ without needing the actual probabilities.

    1. Write the given ratios in terms of $\lambda$.

    $\frac{P(X=5)}{P(X=2)} = \frac{\frac{e^{-\lambda}\lambda^5}{5!}}{\frac{e^{-\lambda}\lambda^2}{2!}} = \frac{\lambda^5 / 120}{\lambda^2 / 2} = \frac{\lambda^3}{60}$

    The problem states this equals $\frac{1}{7500}$. So:

    $\frac{\lambda^3}{60} = \frac{1}{7500}$

    2. Solve for $\lambda^3$ from the first ratio. Multiply both sides by 60:

    $\lambda^3 = \frac{60}{7500} = \frac{1}{125}$

    Hence $\lambda = \sqrt[3]{\frac{1}{125}} = \frac{1}{5}$.

    3. Check consistency with the second ratio.

    $\frac{P(X=5)}{P(X=3)} = \frac{\lambda^5 / 120}{\lambda^3 / 6} = \frac{\lambda^2}{20}$

    Plug $\lambda = 1/5$: …

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