Algebra

Probability

Medium

applied

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

1/4

1/8

3/4

1/3

✅ Correct Option: 1

For a binomial distribution with 3 trials, the probability of exactly $k$ successes is:

$P(X = k) = C(n,k) \times p^k \times (1-p)^{n-k}$

where $n = 3$ is the number of trials and $p$ is the probability of success in each trial.

For 2 successes in 3 trials:

$P(X=2) = C(3,2) \times p^2 \times (1-p)^1$

$C(3,2) = \dfrac{3!}{2! \times 1!} = 3$

$P(X=2) = 3p^2(1-p)$

For 3 successes in 3 trials:

$P(X=3) = C(3,3) \times p^3 \times (1-p)^0$

$C(3,3) = 1$

$(1-p)^0 = 1$

$P(X=3) = p^3$

Given that the probability of 2 successes is 9 times the probability of 3 successes:

$P(X=2) = 9 \times P(X=3)$

$3p^2(1-p) = 9p^3$

$3p^2 - 3p^3 = 9p^3$

$3p^2 = 12p^3$

Dividing both sides by $3p^2$ (where $p \neq 0$ ):

$1 = 4p$

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

Therefore, the probability of success in each trial is $\dfrac{1}{4}$ .

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:

2025: 16 May Shift 1

In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown. The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3. If X denotes the amount which the man wins or loses, then which of the following are correct?

(A) All the possible values of X are 8, 5, 2 and -1.

(B) The probability distribution of X is:

X	8	5	2	-1	-12
P(X)	1/2	1/4	1/8	1/16	1/16

(D) The variance of X is 6615/256.

Choose the correct answer from the options given below:

2026: 30 May Shift 2

A random variable X has the following probability distribution:

X	0	1	2	3	4	5	6
P(X)	z	2z	2z	3z	$z^2$	$2z^2$	$7z^2+z$

Which of the following are TRUE?

(A) P(X < 3) = 0.8

(B) P(X > 4) = 0.2

(D) $P(3\le X<5) = 0.31$

Choose the correct answer from the options given below:

2025: 22 May Shift 2

The random variable X can take values 0, 1, 2. If $P(X=0)=P(X=1)=\alpha$ , and $E(X^2)=E(X)$ , then which of the following are correct?

(A) $E(X) = 2-3\alpha$

(B) $E(X^2) = 4+7\alpha$

(D) $\alpha = \frac{1}{5}$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?

2022: 16 July Shift 2

Let X be the random variable with probability distribution given by the following table.

X	0	1	2	3
P(X = x)	$\frac{1}{8}$	k	$\frac{3}{8}$	$\frac{1}{8}$

The value of $P(X \leq 1)$ is:

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

Solution

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