Algebra

Probability

Medium

core

A committee of 4 students is selected at random from a group consisting of 8 girls and 4 boys. Given that there is at least one boy in the committee, the probability that there are exactly 2 boys in the committee is

$\dfrac{85}{99}$

$\dfrac{168}{425}$

$\dfrac{56}{165}$

$\dfrac{257}{425}$

✅ Correct Option: 2

Total committees $= \binom{12}{4} = 495$ ; those with no boy $= \binom{8}{4} = 70$ .

So committees with at least one boy $= 495-70 = 425$ .

Committees with exactly two boys $= \binom{4}{2}\binom{8}{2} = 6\times28 = 168$ .

Required conditional probability $= \dfrac{168}{425}$ .

2025: 30 May Shift 2

Mean and variance of a binomial distribution are 6 and 2 respectively. The probability of 2 successes will be

2026: 14 May Shift 2

A bag contains 5 white, 4 black and 3 red balls. Which of the following statements are True?

A. The probability of drawing three red balls one by one without replacement is $\frac{1}{220}$

B. One by one three balls drawn without replacement then probability that third ball is red is $\frac{1}{4}$

C. The probability of drawing 2 white balls one by one without replacement is $\frac{3}{4}$

D. One by one 3 balls are drawn without replacement then probability that third ball is red is $\frac{3}{4}$

Choose the correct answer from the options given below:

2026: 15 May Shift 1

Let A and B are two events associated to a random experiment, then

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $P(A \cap B) + P(A \cap \overline{B})$	I. $P(A \cup B)$
B. $P(A \cap B) + P(\overline{A} \cap B)$	II. $P(A) + P(B)$
C. $P(A \cap B) + P(\overline{A} \cap B) + P(A \cap \overline{B})$	III. $P(A)$
D. $P(A \cup B) + P(A \cap B)$	IV. $P(B)$

Choose the correct answer from the options given below:

2025: 22 May Shift 1

A random variable X has the following probability distribution:

X	-2	-1	0	1	2	3
P(X)	0.1	0.2	k	0.3	2k	0.1

then which of the following are TRUE?

(A) $k=0.1$

(B) $P(X < 1) = 0.4$

(D) $P(0 < X < 3) = 0.5$

Choose the correct answer from the options given below:

2026: 12 May Shift 2

From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. Let X denotes the number of defective items in the sample. If the sample is drawn randomly, then Match List-I with List-II (P(X) denotes the probability of occurence of X)

List-I	List-II
(A) P (X = 0)	(I) $\dfrac{^3C_3\times{}^7C_1}{^{10}C_4}$
(B) P (X = 1)	(II) $\dfrac{^7C_4\times{}^3C_0}{^{10}C_4}$
(C) P (X = 2)	(III) $\dfrac{^3C_1\times{}^7C_3}{^{10}C_4}$
(D) P (X = 3)	(IV) $\dfrac{^3C_2\times{}^7C_2}{^{10}C_4}$

Choose the correct answer from the options given below:

2024: 16 May Shift 1

There are $6$ cards numbered $1$ to $6$ , one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then $\mathrm{P}(\mathrm{X}>3)$ is :

A committee of 4 students is selected at random from a group consisting of 8 girls and 4 boys. Given that there is at least one boy in the committee, the probability that there are exactly 2 boys in the committee is

Solution

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