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Probability Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Probability Set-4

Practice Important MCQs for Probability — Mathematics, Class 12.

Probability is a core chapter in Class 12 Mathematics because it trains you to model real-life uncertainty using conditional probability, counting principles, and results on independence. These skills directly appear in board exams and are heavily used in competitive exams to test reasoning through Bayes’ theorem, conditional events, and “given that…” statements.

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Q1. Two fair six-faced dice are rolled. Given that at least one die shows $4$ , what is the probability that the sum of the two faces is $7$ ?

(A)

$\dfrac{1}{6}$

(B)

$\dfrac{1}{5}$

(C)

$\dfrac{2}{11}$

(D)

$\dfrac{1}{3}$

Q2. Machines $M_1,M_2,M_3$ produce $50\%$ , $30\%$ , $20\%$ of the items respectively. Their defective rates are $2\%$ , $3\%$ , $5\%$ respectively. If a randomly chosen item is found defective, the probability it was produced by $M_3$ is

(A)

$\dfrac{10}{29}$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{10}{31}$

(D)

$\dfrac{2}{5}$

Q3. Three fair dice are rolled. Given that at least one die shows a $6$ , what is the probability that exactly two dice show the same face (i.e., there is exactly one pair and the third die shows a different number)?

(A)

$\dfrac{10}{33}$

(B)

$\dfrac{30}{91}$

(C)

$\dfrac{1}{3}$

(D)

$\dfrac{15}{91}$

Q4. Assertion (A): If three events $A,B,C$ are pairwise independent then they are necessarily mutually independent.
Reason (R): Mutual independence requires $P(A\cap B\cap C)=P(A)P(B)P(C)$ in addition to pairwise independence; pairwise independence alone need not imply this.

(A)

Both (A) and (R) are true and (R) is a correct explanation of (A).

(B)

Both (A) and (R) are true but (R) is not a correct explanation of (A).

(C)

(A) is true but (R) is false.

(D)

(A) is false but (R) is true.

Q5. Assertion (A): Let $A$ and $B$ be events with $P(A)>0,P(B)>0$ . If $P(A\mid B)>P(A)$ then $P(B\mid A)>P(B)$ .
Reason (R): From $P(A\mid B)>P(A)$ we get $P(A\cap B)>P(A)P(B)$ ; dividing both sides by $P(A)$ yields $P(B\mid A)>P(B)$ .

(A)

Both (A) and (R) are true and (R) is a correct explanation of (A).

(B)

Both (A) and (R) are true but (R) is not a correct explanation of (A).

(C)

(A) is true but (R) is false.

(D)

(A) is false but (R) is true.

Q6. A bag contains $5$ white, $4$ black and $3$ red balls. Two balls are drawn at random without replacement. Given that at least one of the drawn balls is white, what is the probability that both drawn balls are white?

(A)

$\dfrac{5}{33}$

(B)

$\dfrac{15}{22}$

(C)

$\dfrac{2}{9}$

(D)

$\dfrac{1}{3}$

Q7. A number is chosen uniformly at random from the set $\{1,2,\dots,100\}$ . Given that the chosen number is not divisible by $2$ or $3$ , what is the probability that it is divisible by $5$ ?

(A)

$\dfrac{7}{33}$

(B)

$\dfrac{7}{100}$

(C)

$\dfrac{1}{5}$

(D)

$\dfrac{2}{9}$

Q8. Three cards are drawn at random without replacement from a standard deck of $52$ cards. Given that at least one ace is among the three drawn cards, what is the probability that exactly two of the drawn cards are aces?

(A)

$\dfrac{288}{22100}$

(B)

$\dfrac{4804}{22100}$

(C)

$\dfrac{72}{22100}$

(D)

$\dfrac{72}{1201}$

Q9. Let $X$ be a continuous random variable with density $f(x)=ax^{2}$ for $0\le x\le 2$ and $f(x)=0$ otherwise. Find $P\!\left(\tfrac{1}{2}<X<1 \mid X>\tfrac{1}{3}\right)$ .

(A)

$\dfrac{7}{64}$

(B)

$\dfrac{189}{1720}$

(C)

$\dfrac{3}{10}$

(D)

$\dfrac{7}{215}$

Q10. A well-shuffled standard deck of $52$ cards is turned up one card at a time until the first ace appears. What is the expected number of cards revealed?

(A)

$\dfrac{52}{4}$

(B)

$\dfrac{53}{4}$

(C)

$\dfrac{53}{5}$

(D)

$11$

Q11. Two fair six-faced dice are rolled once each. Given that at least one die shows $4$ , what is the probability that the sum of the two dice is $9$ ?

(A)

$\dfrac{1}{6}$

(B)

$\dfrac{1}{11}$

(C)

$\dfrac{2}{11}$

(D)

$\dfrac{1}{3}$

Q12. Three independent coins are tossed once: coin 1 has $P(H)=\tfrac{1}{2}$ , coin 2 has $P(H)=0.6$ , coin 3 has $P(H)=p$ (with $0<p<1$ ). Given that exactly two heads appear, the probability that coin 1 showed head is

(A)

$\dfrac{3-p}{3+2p}$

(B)

$\dfrac{3-p}{3+p}$

(C)

$\dfrac{1-p}{1+p}$

(D)

$\dfrac{2-p}{3+2p}$

Q13. Let $X$ be the number of successes in $n$ independent Bernoulli trials with success probability $p$ ( $0<p<1$ , $n\ge 1$ ). Given that $X$ is even, the conditional probability $P(X=0\mid X\text{ is even})$ equals

(A)

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

(B)

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

(C)

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

(D)

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

Q14. From $5$ married couples (total $10$ persons), $4$ persons are chosen at random. What is the probability that no selected pair consists of a married couple?

(A)

$\dfrac{4}{21}$

(B)

$\dfrac{8}{21}$

(C)

$\dfrac{16}{35}$

(D)

$\dfrac{2}{7}$

Q15. An urn contains $3$ red, $3$ blue and $3$ green balls. Balls are drawn one by one without replacement. What is the probability that balls of all three colours have appeared for the first time exactly on the $4$ th draw?

(A)

$\dfrac{9}{28}$

(B)

$\dfrac{3}{8}$

(C)

$\dfrac{1}{3}$

(D)

$\dfrac{7}{24}$

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