Random Variables and Probability Distributions

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JEE Main 2011

Consider 5 independent Bernoulli's trials each with probability of success $p$ . If the probability of at least one failure is greater than or equal to $31/32$ , then $p$ lies in the interval

(3/4, 11/12]

(B)

[0, 1/2]

(C)

(11/12, 1]

(D)

[1/2, 3/4]

JEE Main 2006

LEVELBoard

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

(A)

6/ 5^{e}

A random variable $X$ has Poisson distribution with mean 2. Then $P (X > 1.5)$ equals

(A)

2/ e^{2}

A random variable $X$ has the probability distribution: | $X$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $P (X)$ | 0.15 | 0.23 | 0.12 | 0.10 | 0.20 | 0.08 | 0.07 | 0.05 | For the events $E = {X is a prime number}$ and $F = {X < 4}$ , the $P (E \cup F)$ is

The probability of India winning a test match against West Indies is $1/2$ . Assuming independence from match to match the probability that in a $5$ match series India's second win occurs at third test is

(A)

1/8

(B)

1/4

(C)

1/2

(D)

2/3

JEE Advanced 1984

LEVELJEE Main

A box contains $24$ identical balls of which $12$ are white and $12$ are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $4$ th time on the $7$ th draw is

(A)

5/64

(B)

27/32

(C)

5/32

(D)

1/2

JEE Advanced 2015

LEVELJEE Main

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is .........

JEE Advanced 2009

LEVELJEE Main

Comprehension Passage

A fair die is tossed repeatedly until a six is obtained. Let

X

denote the number of tosses required.

Question 1:

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:

(A)

\frac{12}{5}

(B)

6

(C)

4

(D)

\frac{6}{25}

JEE Main 2019 (9 January)

LEVELJEE Main

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then $P (X = 1) + P (X = 2)$ equals :

JEE Main 2019 (12 January)

LEVELJEE Main

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

(A)

400/3

gain

(B)

400/3

loss

(C)

0

(D)

400/9

loss

JEE Main 2019 (12 April)

LEVELBoard

A person throws two fair dice. He wins Rs. 15 for throwing a doublet, wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

JEE Main 2020 (7 January Shift 1)

LEVELJEE Main

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$ , otherwise $X$ takes the value $- 1$ . The expected value of $X$ , is

(A)

\frac{1}{8}

(B)

\frac{3}{16}

(C)

- \frac{1}{8}

(D)

- \frac{3}{16}

JEE Main 2020 (9 January Shift 2)

LEVELJEE Main

A random variable $X$ has the following probability distribution: X: 1, 2, 3, 4, 5; P(X): $K^{2}, 2 K, K, 2 K, 5 K^{2}$ . Then $P (X > 2)$ is equal to:

(A)

\frac{7}{12}

(B)

\frac{23}{36}

(C)

\frac{1}{36}

(D)

\frac{1}{6}

JEE Main 2020 - 7 Jan (Evening)

LEVELJEE Main

In a workshop there are five machines and the probability of any one of them to be out of service on a day is $1/4$ . If the probability that at most two machines will be out of service on the same day is $(3/4)^{3} * k$ then $k$ is equal to:

JEE Main 2020 - 9 Jan (Evening)

LEVELJEE Main

A random variable $X$ has the following probability distribution: $X : 1, 2, 3, 4, 5$ $P (x) : K^{2}, 2 k, K, 2 K, 5 K^{2}$ Then $P (x > 2)$ is equal to :

JEE Main 2020 - 4 Sep (Morning)

LEVELBoard

The probability of a man hitting a target is $1/10$ . The least number of shots required, so that the probability of his hitting the target at least once is greater than $1/4$ , is

JEE Main 2020 - 5 Sep (Morning)

LEVELJEE Main

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is

JEE Main 2020 - 5 Sep (Evening)

LEVELJEE Main

In a bombing attack, there is 50% chance that a bomb will hit the target. Atleast two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is

JEE Main 2020 - 7 Jan (Morning)

LEVELJEE Main

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$ otherwise $X$ takes the value $- 1$ . Then the expected value of $X$ , is:

(A)

1/8

(B)

- 1/8

(C)

3/8

(D)

- 3/8

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Main

Let $X$ be a random variable with distribution. $x P (X = x) - 2 \frac{1}{5} - 1 a 3 \frac{1}{3} 4 \frac{1}{5} 6 b$ If the mean of $X$ is 2.3 and variance of $X$ is $σ^{2}$ , then $100 σ^{2}$ is equal to :

JEE Main 2021 (27 Aug Shift 2)

LEVELBoard

The probability distribution of random variable $X$ is given by: $X P (X) 1 K 2 2 K 3 2 K 4 3 K 5 K$ Let $p = P (1 < X < 4∣ X < 3)$ . If $5 p = λ K$ , then $λ$ equal to .

JEE Main 2021 (27 Aug Shift 2)

LEVELBoard

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

(A)

\frac{1}{8}

(B)

\frac{5}{8}

(C)

\frac{5}{16}

(D)

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P (X \geq 5∣ X > 2)$ is :

(A)

\frac{125}{216}

(B)

\frac{11}{36}

(C)

\frac{5}{6}

(D)

\frac{25}{36}

JEE Main 2021 (25 July Shift 2)

LEVELJEE Main

Let $X$ be a random variable such that the probability function of a distribution is given by $P (X = 0) = 1/2, P (X = j) = 1/ 3^{j} (j = 1, 2, 3, \dots, \infty)$ . Then the mean of the distribution and $P (X is positive and even)$ respectively are:

(A)

3/8

and

1/8

(B)

3/4

and

1/8

(C)

3/4

and

1/9

(D)

3/4

and

1/16

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A random variable X has Poisson distribution with mean 2. Then P(X>1.5) equals

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is .........

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that X=3 equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that X≥3 equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The conditional probability that X≥6 given X>3 equals

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

A person throws two fair dice. He wins Rs. 15 for throwing a doublet, wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A random variable X has the following probability distribution: X: 1, 2, 3, 4, 5; P(X): K2,2K,K,2K,5K2. Then P(X>2) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A random variable X has the following probability distribution: X:1,2,3,4,5 P(x):K2,2k,K,2K,5K2 Then P(x>2) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability of a man hitting a target is 1/10. The least number of shots required, so that the probability of his hitting the target at least once is greater than 1/4, is

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is

In a bombing attack, there is 50% chance that a bomb will hit the target. Atleast two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability distribution of random variable X is given by: XP(X)​1K​22K​32K​43K​5K​​ Let p=P(1<X<4∣X<3). If 5p=λK, then λ equal to .

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A fair die is tossed until six is obtained on it. Let X be the number of required tosses, then the conditional probability P(X≥5∣X>2) is :

A random variable $X$ has Poisson distribution with mean 2. Then $P (X > 1.5)$ equals

The probability that $X = 3$ equals

The probability that $X \geq 3$ equals

The conditional probability that $X \geq 6$ given $X > 3$ equals

A random variable $X$ has the following probability distribution: X: 1, 2, 3, 4, 5; P(X): $K^{2}, 2 K, K, 2 K, 5 K^{2}$ . Then $P (X > 2)$ is equal to:

A random variable $X$ has the following probability distribution: $X : 1, 2, 3, 4, 5$ $P (x) : K^{2}, 2 k, K, 2 K, 5 K^{2}$ Then $P (x > 2)$ is equal to :

The probability of a man hitting a target is $1/10$ . The least number of shots required, so that the probability of his hitting the target at least once is greater than $1/4$ , is

The probability distribution of random variable $X$ is given by: $X P (X) 1 K 2 2 K 3 2 K 4 3 K 5 K$ Let $p = P (1 < X < 4∣ X < 3)$ . If $5 p = λ K$ , then $λ$ equal to .

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P (X \geq 5∣ X > 2)$ is :