Advertisements
Advertisements
प्रश्न
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
उत्तर
(a) Chetan has no black hair or no blue eyes.
(b) ∀ x ∈ R such that x2 + 3 ≤ 0.
APPEARS IN
संबंधित प्रश्न
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. What are the possible values of X? Is X a random variable?
A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
The probability distribution function of a random variable X is given by
xi : | 0 | 1 | 2 |
pi : | 3c3 | 4c − 10c2 | 5c-1 |
where c > 0 Find: c
Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls
Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.
Find the mean and standard deviation of each of the following probability distribution :
xi: | 0 | 1 | 3 | 5 |
pi : | 0.2 | 0.5 | 0.2 | 0.1 |
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X a random variable ? If yes, then find the mean and variance of X.
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is
A random variable has the following probability distribution:
X = xi : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
P (X = xi) : | 0 | 2 p | 2 p | 3 p | p2 | 2 p2 | 7 p2 | 2 p |
The value of p is
If X is a random-variable with probability distribution as given below:
X = xi : | 0 | 1 | 2 | 3 |
P (X = xi) : | k | 3 k | 3 k | k |
The value of k and its variance are
From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
A random variable X has the following probability distribution :
X = x | -2 | -1 | 0 | 1 | 2 | 3 |
P(x) | 0.1 | k | 0.2 | 2k | 0.3 | k |
Find the value of k and calculate mean.
If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
If X ∼ N (4,25), then find P(x ≤ 4)
From the following data, find the crude death rates (C.D.R.) for Town I and Town II, and comment on the results :
Age Group (in years) | Town I | Town II | ||
Population | No. of deaths | Population | No. of deaths | |
0-10 | 1500 | 45 | 6000 | 150 |
10-25 | 5000 | 30 | 6000 | 40 |
25 - 45 | 3000 | 15 | 5000 | 20 |
45 & above | 500 | 22 | 3000 | 54 |
Determine whether each of the following is a probability distribution. Give reasons for your answer.
x | 0 | 1 | 2 | 3 | 4 |
P(x) | 0.1 | 0.5 | 0.2 | –0.1 | 0.3 |
Determine whether each of the following is a probability distribution. Give reasons for your answer.
x | 0 | 1 | 2 |
P(x) | 0.3 | 0.4 | 0.2 |
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of X ≤ 1
The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of X > 1
Solve the following problem :
The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.
Solve the following problem :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Calculate the probabilities of obtaining an answer yes from all of the selected students.
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.
The probability distribution of a random variable X is given below:
X | 0 | 1 | 2 | 3 |
P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Determine the value of k.
The probability distribution of a random variable X is given below:
X | 0 | 1 | 2 | 3 |
P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Determine P(X ≤ 2) and P(X > 2)
The probability distribution of a random variable X is given below:
X | 0 | 1 | 2 | 3 |
P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Find P(X ≤ 2) + P (X > 2)
Two biased dice are thrown together. For the first die P(6) = `1/2`, the other scores being equally likely while for the second die, P(1) = `2/5` and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.
Two probability distributions of the discrete random variable X and Y are given below.
X | 0 | 1 | 2 | 3 |
P(X) | `1/5` | `2/5` | `1/5` | `1/5` |
Y | 0 | 1 | 2 | 3 |
P(Y) | `1/5` | `3/10` | `2/10` | `1/10` |
Prove that E(Y2) = 2E(X).
A person throws two fair dice. He wins ₹ 15 for throwing a doublet (same numbers on the two dice), wins ₹ 12 when the throw results in the sum of 9, and loses ₹ 6 for any other outcome on the throw. Then the expected gain/loss (in ₹) of the person is ______.
A random variable X has the following probability distribution:
x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
P(x) | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Find:
- k
- P(X < 3)
- P(X > 4)
The probability that a bomb will hit the target is 0.8. Complete the following activity to find, the probability that, out of 5 bombs exactly 2 will miss the target.
Solution: Here, n = 5, X =number of bombs that hit the target
p = probability that bomb will hit the target = `square`
∴ q = 1 - p = `square`
Here, `X∼B(5,4/5)`
∴ P(X = x) = `""^"n""C"_x"P"^x"q"^("n" - x) = square`
P[Exactly 2 bombs will miss the target] = P[Exactly 3 bombs will hit the target]
= P(X = 3)
=`""^5"C"_3(4/5)^3(1/5)^2=10(4/5)^3(1/5)^2`
∴ P(X = 3) = `square`
Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and –1 for any other number.
- If X denotes the random variable “points earned” then what are the possible values X can take?
- Find the probability distribution of this random variable X.
- Find the expected value of the points she gets.