Q.Suppose that $5\%$ of men and $0.25\%$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Showing the 12 most recent of 30 on this concept.

1. 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:

   $P(B_2 \mid A) = \frac{P(B_2) \, P(A \mid B_2)}{P(A)}$

   Plug in the numbers:

   $P(B_2 \mid A) = \frac{0.30 \times 0.04}{0.0380} = \frac{0.0120}{0.0380}$

   4. Simplify the fraction Divide numerator and denominator by 0.002 (or multiply by 1000 to clear decimals):

   $\frac{0.0120}{0.0380} = \frac{12}{38} = \frac{6}{19}$

   So $P(B_2 \mid A) = \frac{6}{19}$.

   Tip

   A common pitfall is forgetting to compute $P(A)$ correctly — students sometimes use only the numerator. Always check that the denominator is the total probability of A, not just one term.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

2. 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 $\lambda = 1/5$:

   $\frac{(1/5)^2}{20} = \frac{1/25}{20} = \frac{1}{500}$

   This matches the given $\frac{1}{500}$, confirming our solution.

   Watch out

   A common mistake is forgetting to include the factorial terms when simplifying the ratio. Always write out $k!$ explicitly: $5! = 120$, $2! = 2$, $3! = 6$.

   Tip

   Notice that the two ratios give two equations, but they are not independent — both lead to the same $\lambda$. You only need one ratio to solve; the second serves as a verification.

   ✓Final answer

   The correct option is (B).

   ANSWER: B

3. 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).

   Watch out

   A common mistake is to forget that $\lambda$ must be positive. Also, do not cancel $\lambda^3$ if $\lambda = 0$ — but $\lambda = 0$ would make all probabilities zero, which contradicts the given relation (since $P(X=3)$ and $P(X=5)$ would both be zero, making $5 \cdot 0 = 0$ trivially true, but that degenerate case is not intended in such problems). Always check that the solution makes physical sense.

   4. Now compute $P(X=2)$ with $\lambda = 10$:

   $P(X=2) = \frac{e^{-10} \cdot 10^2}{2!} = \frac{e^{-10} \cdot 100}{2} = 50 e^{-10}$

   5. Write $e^{-10}$ as $\frac{1}{e^{10}}$, so:

   $P(X=2) = \frac{50}{e^{10}}$

   Tip

   Notice that the answer choices are all of the form $\frac{\text{integer}}{e^{\text{integer}}}$. Once you find $\lambda = 10$, the denominator $e^{10}$ immediately points to option (B). This can be a quick sanity check.

   ✓Final answer

   The value is $\boxed{\frac{50}{e^{10}}}$, which corresponds to option (B).

4. 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 $P(B)$:

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

   Bring terms together:

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

   $\frac{2}{5}P(B) = \frac{3}{20}$

   So:

   $P(B) = \frac{3}{20} \cdot \frac{5}{2} = \frac{15}{40} = \frac{3}{8}$

   3. Compute the required sum

   $P(A \cap B) + P(B) = \frac{3}{20} + \frac{3}{8} = \frac{6}{40} + \frac{15}{40} = \frac{21}{40}$

   Tip

   Notice that $P(A) = P(B) = \frac{3}{8}$ here — a nice symmetry that emerges from the given numbers.

   Watch out

   A common mistake is to treat $P(\overline{A} \mid B)$ as $1 - P(A \mid B)$ incorrectly — that works only if you adjust carefully. Always go back to the definition $P(\overline{A} \mid B) = \frac{P(\overline{A} \cap B)}{P(B)}$.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

5. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
   Q.A non-zero integer $x$ is selected randomly from the set of integers $\{x \in \mathbb{Z} / -25 \leq x \leq 25,\, x \neq 0\}$. The probability that $x + 6 \leq \dfrac{135}{x}$ is (A) $\dfrac{12}{25}$ (B) $\dfrac{2}{5}$ (C) $\dfrac{3}{5}$ (D) $\dfrac{14}{25}$

   ›Reveal solutionSolution

   We need to find the probability that a non-zero integer $x$ from the set $\{-25, \dots, 25\}$ satisfies the inequality $x + 6 \leq \frac{135}{x}$. We first determine the total number of possible integers (the sample space), which is 50. Then, we solve the inequality to find the integers that satisfy it within the given range (the event space), which are 20 integers. The probability is $\boxed{\frac{2}{5}}$.

   The problem asks for a probability, which is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. Our strategy will be to first identify the complete set of possible integers $x$ (the sample space) and count them. Then, we will solve the given inequality to find which of these integers satisfy the condition (the event space) and count those. Finally, we will compute the ratio.

   1. Determine the Sample Space:

      The problem states that $x$ is a non-zero integer selected from the set $\{x \in \mathbb{Z} / -25 \leq x \leq 25,\, x \neq 0\}$.

      This means $x$ can be any integer from $-25$ to $25$, but $x$ cannot be $0$.

      The integers in this range are $\{-25, -24, \dots, -1, 0, 1, \dots, 24, 25\}$.

      The total count of integers from $-25$ to $25$ (inclusive) is $25 - (-25) + 1 = 51$.

      Since $x \neq 0$, we must exclude $0$ from this count.

      Therefore, the total number of possible outcomes (the size of the sample space) is $51 - 1 = 50$.

   2. Solve the Inequality:

      We need to find the integers $x$ that satisfy the inequality $x + 6 \leq \dfrac{135}{x}$.

      To solve rational inequalities, the most reliable method is to move all terms to one side and combine them into a single fraction. This avoids potential errors that arise from multiplying by a variable whose sign is unknown.

      $x + 6 - \dfrac{135}{x} \leq 0$

      To combine these terms, we find a common denominator, which is $x$:

      $\dfrac{x \cdot x}{x} + \dfrac{6 \cdot x}{x} - \dfrac{135}{x} \leq 0$

      $\dfrac{x^2 + 6x - 135}{x} \leq 0$

   3. Factor the Numerator:

      Now, we need to find the roots of the quadratic expression in the numerator, $x^2 + 6x - 135 = 0$. We can use the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

      $x = \dfrac{-6 \pm \sqrt{6^2 - 4(1)(-135)}}{2(1)}$

      $x = \dfrac{-6 \pm \sqrt{36 + 540}}{2}$

      $x = \dfrac{-6 \pm \sqrt{576}}{2}$

      Recognizing that $24^2 = 576$, we have:

      $x = \dfrac{-6 \pm 24}{2}$

      This gives two roots:

      $x_1 = \dfrac{-6 - 24}{2} = \dfrac{-30}{2} = -15$

      $x_2 = \dfrac{-6 + 24}{2} = \dfrac{18}{2} = 9$

      So, the numerator can be factored as $(x - (-15))(x - 9) = (x+15)(x-9)$.

      The inequality now becomes $\dfrac{(x+15)(x-9)}{x} \leq 0$.

      Watch out

      A common mistake is to multiply both sides of the inequality by $x$. This is incorrect because the sign of $x$ is unknown. If $x$ is negative, multiplying by $x$ would reverse the inequality sign. If $x$ is positive, it would not. Handling these two cases separately is cumbersome and prone to error. The method of moving all terms to one side and analyzing critical points is more robust.

   4. Determine Intervals Satisfying the Inequality:

      The critical points are the values of $x$ where the numerator or the denominator is zero. These are $x = -15$, $x = 0$, and $x = 9$. These points divide the number line into four intervals. We will test a value from each interval to determine the sign of the expression $\dfrac{(x+15)(x-9)}{x}$.

      Interval	Test Value ($x$)	Sign of $(x+15)$	Sign of $(x-9)$	Sign of $x$	Sign of $\dfrac{(x+15)(x-9)}{x}$	Condition $\leq 0$
      $x < -15$	$-20$	Negative	Negative	Negative	$\dfrac{(-)(-) }{(-)} = (-)$	True
      $-15 < x < 0$	$-1$	Positive	Negative	Negative	$\dfrac{(+)(-) }{(-)} = (+)$	False
      $0 < x < 9$	$1$	Positive	Negative	Positive	$\dfrac{(+)(-) }{(+)} = (-)$	True
      $x > 9$	$10$	Positive	Positive	Positive	$\dfrac{(+)(+) }{(+)} = (+)$	False

      The inequality $\dfrac{(x+15)(x-9)}{x} \leq 0$ is satisfied when the expression is negative or zero.

      The expression is zero when $x = -15$ or $x = 9$.

      The expression is undefined when $x = 0$.

      Combining these, the solution set for the inequality is $x \in (-\infty, -15] \cup (0, 9]$.

   5. Identify Favorable Outcomes within the Sample Space:

      We need to find the integers that are both in our sample space $\{-25, \dots, -1, 1, \dots, 25\}$ AND satisfy $x \in (-\infty, -15] \cup (0, 9]$.

      • For the interval $x \in (-\infty, -15]$:

        The integers from the sample space that fall into this interval are $\{-25, -24, \dots, -16, -15\}$.

        The number of such integers is $-15 - (-25) + 1 = -15 + 25 + 1 = 11$.

      • For the interval $x \in (0, 9]$:

        The integers from the sample space that fall into this interval are $\{1, 2, \dots, 8, 9\}$.

        The number of such integers is $9 - 1 + 1 = 9$.

      The total number of favorable outcomes (integers satisfying both conditions) is $11 + 9 = 20$.

   6. Calculate the Probability:

      The probability $P$ is the ratio of the number of favorable outcomes to the total number of possible outcomes.

      $P = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \dfrac{20}{50}$

      $P = \dfrac{2}{5}$

      Tip

      Always simplify fractions to their lowest terms.

   ✓Final answer

   The probability that $x + 6 \leq \dfrac{135}{x}$ is $\boxed{\dfrac{2}{5}}$.

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.
      • $\frac{15}{52}$ matches option (A).
      • For completeness: $\frac{4}{13} = \frac{16}{52}$, $\frac{17}{52}$, and $\frac{5}{13} = \frac{20}{52}$ are all different.

   Tip

   A quick sanity check: The probability of exactly one is also $P(A) + P(B) - 2P(A \cap B)$. Many students mistakenly use $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, which gives "at least one" instead of "exactly one."

   Watch out

   A common pitfall is forgetting to subtract the intersection twice. If you only subtract it once, you get $\frac{16}{52} = \frac{4}{13}$, which is option (B) — the probability of at least one, not exactly one.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

7. 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. The denominators are 8, 1, and 10. The LCM is 40. So:

   $\frac{3}{8} = \frac{15}{40},\quad 1 = \frac{40}{40},\quad \frac{3}{10} = \frac{12}{40}.$

   Therefore:

   $P(A \cup B) = \left( \frac{15}{40} + \frac{40}{40} - \frac{12}{40} \right) P(B) = \frac{43}{40} P(B).$

   6. Compare with the given relation $P(A \cup B) = K P(B)$. We have $P(A \cup B) = \frac{43}{40} P(B)$, so $K = \frac{43}{40}$. Hence

   $\frac{1}{K} = \frac{40}{43}.$

   Watch out

   A common mistake is to treat $P(A/B)$ and $P(B/A)$ as if they were the same, or to forget that $P(A \cap B)$ appears in both but must be expressed consistently. Always write the definition first.

   ✓Final answer

   The value is $\boxed{\frac{40}{43}}$, which corresponds to option (B).

8. TG EAPCET 2024Set eng-2024-05-09-FN1 markMCQ
   Q.If the probability that a student selected at random from a particular college is good at mathematics is 0.6, then the probability of having two students who are good at mathematics in a group of 8 students of that college standing in front of the college is (A) $\dfrac{2^6 \times 3^2 \times 7}{5^8}$ (B) $\dfrac{2^6 \times 3^2 \times 7}{5^6}$ (C) $\dfrac{2^8 \times 3^2 \times 7}{5^6}$ (D) $\dfrac{2^8 \times 3^2 \times 7}{5^8}$

   ›Reveal solutionSolution

   This is a binomial trial with $n=8$, $p=0.6$. $P(X=2)=\binom{8}{2}(0.6)^2(0.4)^6 = \dfrac{2^{8}\times 3^{2}\times 7}{5^{8}}$, option (D).

   Binomial model

   Each student is independently good at mathematics with probability $p = 0.6 = \dfrac{3}{5}$, so $q = 0.4 = \dfrac{2}{5}$. For $n=8$ students, the number good at mathematics is binomial, and we want exactly two:

   $P(X=2) = \binom{8}{2}\,p^{2}\,q^{6} = \binom{8}{2}\left(\frac{3}{5}\right)^{2}\left(\frac{2}{5}\right)^{6}.$

   Simplify

   $\binom{8}{2}=28,\qquad \left(\frac{3}{5}\right)^{2}=\frac{3^{2}}{5^{2}},\qquad \left(\frac{2}{5}\right)^{6}=\frac{2^{6}}{5^{6}}.$

   $P(X=2) = 28\cdot\frac{3^{2}}{5^{2}}\cdot\frac{2^{6}}{5^{6}} = \frac{28\cdot 3^{2}\cdot 2^{6}}{5^{8}}.$

   Write $28 = 2^{2}\times 7$:

   $= \frac{2^{2}\times 7\times 3^{2}\times 2^{6}}{5^{8}} = \frac{2^{8}\times 3^{2}\times 7}{5^{8}}.$

   ✓Final answer

   $P(\text{exactly two}) = \dfrac{2^{8}\times 3^{2}\times 7}{5^{8}}$. The correct option is (D).

   ANSWER: D

9. TG EAPCET 2022Set eng-2022-07-18-FN1 markMCQ
   Q.Two balls are drawn at random from a bag containing 5 black balls and 3 white balls. If the random variable X denotes the number of white balls drawn, then the mean of X is (A) $\dfrac{1}{2}$ (B) $\dfrac{5}{8}$ (C) $\dfrac{3}{4}$ (D) $\dfrac{3}{8}$

   ›Reveal solutionSolution

   The mean (expected value) of the number of white balls drawn when picking two balls without replacement from 5 black and 3 white balls is $\frac{3}{4}$. The correct option is (C).

   ---

   We are drawing two balls without replacement from a small finite set. The random variable $X$ counts how many white balls appear. The mean (expected value) is just the average number of whites we’d see if we repeated the draw many times.

   Key insight: Instead of listing all outcomes and probabilities, we can use the linearity of expectation. Each ball drawn is like a “mini-experiment”: define an indicator for whether the first ball is white, and another for the second. The expected number of whites is simply the sum of the probabilities that each individual draw yields a white ball. This works even though the draws are dependent — expectation adds regardless.

   ---

   1. Define indicator variables

      Let $I_1 = 1$ if the first ball is white, $0$ otherwise.

      Let $I_2 = 1$ if the second ball is white, $0$ otherwise.

      Then $X = I_1 + I_2$.

   2. Find the probability the first ball is white

      Initially there are 3 white balls out of 8 total.

   $P(I_1 = 1) = \frac{3}{8}.$

   3. Find the probability the second ball is white By symmetry (or by the law of total probability), the chance the second ball is white is also $\frac{3}{8}$. Why? Because without any information about the first draw, the second ball is equally likely to be any of the 8 original balls. So

   $P(I_2 = 1) = \frac{3}{8}.$

   4. Apply linearity of expectation

   $E[X] = E[I_1 + I_2] = E[I_1] + E[I_2] = \frac{3}{8} + \frac{3}{8} = \frac{6}{8} = \frac{3}{4}.$

   Tip

   A common pitfall is to think the second draw’s probability changes because the first draw removed a ball. But without conditioning on the first result, the second draw still has a $\frac{3}{8}$ chance of being white — symmetry saves us.

   Watch out

   Do not compute the distribution of $X$ from scratch unless you enjoy extra work. The direct method (listing $P(X=0), P(X=1), P(X=2)$) gives the same answer but is slower. Here, linearity makes it a one-liner.

   ---

   ✓Final answer

   The correct option is (C).

   ANSWER: C

10. 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:

       • Row 1: $\{1,2,3\}$ → sum = 6 (even) → not in A.
       • Row 2: $\{4,5,6\}$ → sum = 15 (odd) → in A.
       • Row 3: $\{7,8,9\}$ → sum = 24 (even) → not in A.
       • Column 1: $\{1,4,7\}$ → sum = 12 (even) → not in A.
       • Column 2: $\{2,5,8\}$ → sum = 15 (odd) → in A.
       • Column 3: $\{3,6,9\}$ → sum = 18 (even) → not in A. So exactly 2 triples (row 2 and column 2) satisfy both. Hence $|A \cap B| = 2$.

    5. Conditional probability $P(A/B)$

    $P(A/B) = \frac{|A \cap B|}{|B|} = \frac{2}{6} = \frac{1}{3}.$

    6. Add the two probabilities

    $P(A) + P(A/B) = \frac{10}{21} + \frac{1}{3} = \frac{10}{21} + \frac{7}{21} = \frac{17}{21}.$

    Tip

    A common mistake is to compute $P(A/B)$ using the full sample space instead of restricting to B. Always remember: conditional probability uses only the outcomes in B as the denominator.

    ✓Final answer

    The correct option is (B).

    ANSWER: B

11. 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 $\lambda=3$:

       Using the PMF for $x=5$:

       $P(X=5) = \frac{e^{-\lambda} \lambda^5}{5!}$

       Substitute $\lambda=3$:

       $P(X=5) = \frac{e^{-3} 3^5}{5!}$

       Calculate $3^5 = 243$ and $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

       $P(X=5) = \frac{e^{-3} \cdot 243}{120}$

       Simplify the fraction $\frac{243}{120}$ by dividing both numerator and denominator by their greatest common divisor, which is 3:

       $P(X=5) = \frac{81e^{-3}}{40}$

    5. Express $P(X=5)$ in terms of $k$:

       From Step 3, we have $k = \frac{27e^{-3}}{10}$.

       From Step 4, we have $P(X=5) = \frac{81e^{-3}}{40}$.

       We need to find a relationship between $\frac{81e^{-3}}{40}$ and $\frac{27e^{-3}}{10}$.

       Notice that $81 = 3 \times 27$ and $40 = 4 \times 10$.

       So, we can rewrite $P(X=5)$ as:

       $P(X=5) = \frac{3 \times (27e^{-3})}{4 \times 10} = \frac{3}{4} \left( \frac{27e^{-3}}{10} \right)$

       Substitute the expression for $k$:

       $P(X=5) = \frac{3}{4} k$

    6. Compare with the given options:

       The calculated value $P(X=5) = \frac{3}{4}k$ matches option (D).

    ✓Final answer

    The value of $P(X=5)$ is $\boxed{\frac{3}{4}k}$.

12. TG EAPCET 2021Set eng-2021-08-04-FN1 markMCQ
    Q.If the probability that an individual will suffer a bad reaction from an injection is 0.001, then the probability that out of 2000 individuals, exactly 3 individuals suffer a bad reaction is (A) $\dfrac{4}{3e^2}$ (B) $\dfrac{2}{e^2}$ (C) $\dfrac{2}{3e^2}$ (D) $\dfrac{4}{5e^2}$

    ›Reveal solutionSolution

    This is a rare-event problem where $n=2000$ is large and $p=0.001$ is small, so the Poisson approximation to the binomial distribution applies. The probability that exactly 3 individuals suffer a reaction is $\boxed{\dfrac{4}{3e^2}}$, which corresponds to option (A).

    The key insight here is that we are dealing with a binomial experiment: 2000 independent trials, each with a constant probability of "success" (bad reaction) $p = 0.001$. The exact probability of exactly 3 successes would be $\binom{2000}{3} (0.001)^3 (0.999)^{1997}$. That expression is perfectly correct but computationally messy — and more importantly, it misses the conceptual point.

    When $n$ is large and $p$ is very small, the binomial distribution is well approximated by the Poisson distribution with parameter $\lambda = np$. This is the classic "rare event" scenario. The Poisson distribution gives us a clean, closed-form answer that matches one of the given options exactly.

    Let's work through it step by step.

    1. Identify the exact binomial probability. For $n = 2000$, $p = 0.001$, the probability of exactly $k = 3$ reactions is

    $P(X = 3) = \binom{2000}{3} (0.001)^3 (0.999)^{1997}.$

    This is the truth, but we won't evaluate it directly — the numbers are unwieldy and the exam expects the Poisson approximation.

    2. Compute the Poisson parameter $\lambda$.

    $\lambda = np = 2000 \times 0.001 = 2.$

    This $\lambda = 2$ is the average number of bad reactions expected in 2000 individuals.

    3. Apply the Poisson approximation. For a Poisson random variable $Y$ with mean $\lambda$,

    $P(Y = k) = \frac{e^{-\lambda} \lambda^k}{k!}.$

    Here $\lambda = 2$ and $k = 3$, so

    $P(Y = 3) = \frac{e^{-2} \cdot 2^3}{3!} = \frac{e^{-2} \cdot 8}{6} = \frac{8}{6e^2} = \frac{4}{3e^2}.$

    4. Check the match with the options. The result $\frac{4}{3e^2}$ is exactly option (A). The other options differ in the numerator or denominator, so there is no ambiguity.

    Watch out

    A common mistake is to forget the factorial in the denominator. $3! = 6$, not $3$ or $2$. Also, note that $e^{-2}$ is written as $\frac{1}{e^2}$, so the final expression is $\frac{4}{3e^2}$, not $\frac{4}{3e}$ or anything else.

    Tip

    The Poisson approximation works beautifully here because $n$ is large ($2000$) and $p$ is small ($0.001$), with $np = 2$ being moderate. A rule of thumb: if $n \ge 20$ and $p \le 0.05$, the approximation is good; here it's excellent.

    ✓Final answer

    The probability is $\boxed{\dfrac{4}{3e^2}}$, which is option (A).