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Probability Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Probability Set-6

Practice Important MCQs for Probability — Mathematics, Class 12.

Probability is one of the most scoring and concept-heavy units in Class 12 Mathematics and also appears frequently in competitive exams (JEE/NEET). Mastering probability ideas like conditional probability, Bayes’ theorem, random variables/distributions, and counting methods helps you solve both numerical and reasoning-based questions quickly and accurately—especially those testing intuition about “given that…” and dependence/independence.

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Q1. A box contains 5 red and 7 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls drawn are blue?

(A)

$\dfrac{7}{24}$

(B)

$\dfrac{7}{22}$

(C)

$\dfrac{35}{132}$

(D)

$\dfrac{5}{12}$

Q2. Coins $C_1,C_2,C_3$ have probabilities of heads $0.5,\,0.7,\,0.2$ respectively. One coin is chosen at random and tossed twice. Given both tosses show heads, the probability that the chosen coin was $C_2$ equals

(A)

$\dfrac{49}{300}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{7}{11}$

(D)

$\dfrac{49}{78}$

Q3. A 5-card hand is drawn at random from a standard 52-card deck. Given that the hand contains at least one ace, the probability that it contains exactly two aces is

(A)

$\displaystyle \frac{\binom{4}{2}\binom{48}{3}}{\binom{52}{5}-\binom{48}{5}}$

(B)

$\displaystyle \frac{\binom{4}{2}\binom{48}{3}}{\binom{52}{5}}$

(C)

$\displaystyle \frac{\binom{4}{1}\binom{48}{4}}{\binom{52}{5}-\binom{48}{5}}$

(D)

$\displaystyle \frac{\binom{4}{2}\binom{48}{3}}{\binom{52}{5}-\binom{47}{5}}$

Q4. Let $X\sim\mathrm{Bin}(n,p)$ with $0<p<1$ . The probability that $X$ is even equals

(A)

$\dfrac{1-(1-2p)^n}{2}$

(B)

$\dfrac{1-(1-2p)^{n+1}}{2}$

(C)

$\dfrac{1+(1-2p)^n}{2}$

(D)

$\dfrac{1+(1-2p)^{n+1}}{2}$

Q5. Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda>0$ . The conditional density of $X$ given $X<Y$ is

(A)

$\lambda e^{-\lambda x},\quad x\ge 0$

(B)

$2\lambda e^{-2\lambda x},\quad x\ge 0$

(C)

$\lambda e^{-2\lambda x},\quad x\ge 0$

(D)

$\tfrac{1}{2}\lambda e^{-\lambda x},\quad x\ge 0$

Q6. Let $X\sim\operatorname{Bin}(4,p)$ and $P(X=0)=P(X=4)$ . Then

(A)

$p=\dfrac{1}{4}$

(B)

$p=\dfrac{1}{2}$

(C)

$p=\dfrac{3}{4}$

(D)

$p=\dfrac{1}{16}$

Q7. An urn contains $5$ red and $7$ blue balls. Two balls are drawn at random without replacement. Given that at least one of the drawn balls is red, the probability that both drawn balls are red is

(A)

$\dfrac{2}{9}$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{2}{11}$

(D)

$\dfrac{5}{33}$

Q8. Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$ . The quadratic equation $t^2 - Xt + Y = 0$ has real roots iff $X^2-4Y\ge0$ . The probability that the equation has real roots is

(A)

$\dfrac{1}{6}$

(B)

$\dfrac{1}{8}$

(C)

$\dfrac{1}{4}$

(D)

$\dfrac{1}{12}$

Q9. A fair coin is tossed repeatedly until either three heads have appeared or two tails have appeared, whichever occurs first. The probability that the process stops because three heads have appeared is

(A)

$\dfrac{3}{8}$

(B)

$\dfrac{1}{4}$

(C)

$\dfrac{5}{16}$

(D)

$\dfrac{5}{32}$

Q10. An urn contains $m$ white and $n$ black balls with $m,n\ge1$ . Balls are drawn one by one at random without replacement until a white ball appears. Let $T$ denote the number of draws required. Then $E[T]$ equals

(A)

$\dfrac{m+n+1}{m+1}$

(B)

$\dfrac{m+n}{m}$

(C)

$\dfrac{n+1}{m}$

(D)

$\dfrac{n+1}{m+1}$

Q11. A student answers 5 multiple-choice questions by random guessing; each question has 4 options with exactly one correct option. What is the probability that exactly two answers are correct?

(A)

$\dfrac{75}{1024}$

(B)

$\dfrac{135}{512}$

(C)

$\dfrac{27}{128}$

(D)

$\dfrac{5}{32}$

Q12. Three fair dice are thrown simultaneously. What is the probability that the maximum face value is $5$ and the minimum face value is $3$ ?

(A)

$\dfrac{1}{27}$

(B)

$\dfrac{2}{27}$

(C)

$\dfrac{1}{12}$

(D)

$\dfrac{1}{18}$

Q13. Let $X$ be a geometric random variable with parameter $p$ (i.e., $P(X=k)=(1-p)^{k-1}p$ for $k=1,2,\dots$ ). If $P(X\ \text{is odd})=\dfrac{3}{5}$ , find $p$ .

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{2}{5}$

(C)

$\dfrac{1}{2}$

(D)

$\dfrac{1}{4}$

Q14. Two points are chosen independently and uniformly at random from the interval $[0,1]$ . What is the probability that the distance between them is less than $\dfrac{1}{3}$ ?

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{2}{3}$

(C)

$\dfrac{5}{9}$

(D)

$\dfrac{4}{9}$

Q15. Let $X_1$ and $X_2$ be independent random variables with common density $f(x)=2x$ for $0<x<1$ and $0$ otherwise. What is $P(X_1>2X_2)$ ?

(A)

$\dfrac{1}{8}$

(B)

$\dfrac{1}{4}$

(C)

$\dfrac{3}{16}$

(D)

$\dfrac{1}{12}$

...and 5 more challenging questions available in the interactive simulator.

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