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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X? - Mathematics and Statistics

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प्रश्न

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

बेरीज

उत्तर १

The urn contains 5 red and 2 black balls. If two balls are drawn from the urn, it contains either 0 or 1 or 2 black balls.

X can take values 0, 1, 2.

∴ X = {0, 1, 2}.

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उत्तर २

X denotes the number of black balls drawn.

Sample space of the experiment is

S = {RR, BR, RB, BB}

The value of X corresponding to these outcomes are as follows:

X (RR) = 0

X(BR) = X(RB) = 1

X(BB) = 2

∴ Possible values of X are {0, 1, 2}.

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पाठ 7: Probability Distributions - Exercise 7.1 [पृष्ठ २३२]

संबंधित प्रश्‍न

A random variable X has the following probability distribution:

then E(X)=....................


From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -

(a) exactly 1 defective bulb.

(b) at least 1 defective bulb.


Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is hostler?


State the following are not the probability distributions of a random variable. Give reasons for your answer.

X 0 1 2 3 4
P(X) 0.1 0.5 0.2 -0.1 0.3

State the following are not the probability distributions of a random variable. Give reasons for your answer.

Y -1 0 1
P(Y) 0.6 0.1 0.2

Find the probability distribution of number of heads in two tosses of a coin.


Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as

(i) number greater than 4

(ii) six appears on at least one die


From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.


A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.


The random variable X has probability distribution P(X) of the following form, where k is some number:

`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`

  1. Determine the value of 'k'.
  2. Find P(X < 2), P(X ≥ 2), P(X ≤ 2).

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]

 


Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. 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.


Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.


Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.


Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?


A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?


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


From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement .

 

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. 


The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

Determine the value of k .


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 number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}k\text{ x }  & , & \text{ if } x = 0 \text{ or }  1 \\ 2 \text{ kx }  & , & \text{ if }  x = 2 \\ k\left( 5 - x \right) & , & \text{ if } x = 3 \text{ or } 4 \\ 0 & , & \text{ if } x > 4\end{cases}\]

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.


Find the mean and standard deviation of each of the following probability distributions:

xi : 2 3 4
pi : 0.2 0.5 0.3

 


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

Find the mean and standard deviation of each of the following probability distribution :

xi :  -2 -1 0 1 2
pi :  0.1 0.2 0.4 0.2 0.1

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.


A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.


A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.


A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.


A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.


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


In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.

 

Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:

Xxi: -2 -1 0 1
P(Xxi) :
\[\frac{1 - a}{4}\]
 
\[\frac{1 + 2a}{4}\]
\[\frac{1 - 2a}{4}\]
\[\frac{1 + a}{4}\]

A random variable X has the following 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 : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is


Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is                

 

 


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. 


Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribtution. 


Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.


Let X be a random variable which assumes values  x1 , x2, x3 , x4 such that  2P (X = x1) = 3P (X = x2) = P (X = x3) = 5P (X = x4). Find the probability distribution of X.


Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following: 

Age x 0 1 2
lx 1000 880 876
T - - 3323

Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.


Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).


Verify the following function, which can be regarded as p.m.f. for the given values of X : 

X = x -1 0 1
P(x) -0.2 1 0.2

John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them? 


Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows : 

Aeroplanes  Cargo consignments 
C1 C2 C3
A1 1 4 5
A2 2 3 3
A3 3 1 2

How should the cargo consignments be assigned to the aeroplanes to maximize the profit? 


A fair coin is tossed 12 times. Find the probability of getting  at least 2 heads .


If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)


Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ₹900 on conveyance, find his expenditure on food items during the same month. 


The following table gives the age of the husbands and of the wives : 

Age of wives (in years)

Age of husbands (in years)

20-30  30- 40  40- 50  50- 60 
15-25  5 9 3 -
25-35  - 10 25 2
35-45  - 1 12 2
45-55  - - 4 16
55-65  - - - 4

Find the marginal frequency distribution of the age of husbands. 


Verify whether the following function can be regarded as probability mass function (p.m.f.) for the given values of X : 

X -1 0 1
P(X = x) -0.2 1 0.2

The defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have-

(a) No defects
(b) At least one defect 
[Use e-1 = 0.3678]


The probability that a bomb dropped from an aeroplane will strike a target is `1/5`, If four bombs are dropped, find the probability that : 

(a) exactly two will strike the target,
(b) at least one will strike the target. 


A random variable X has the following probability distribution :

X 0 1 2 3 4 5 6
P(X) C 2C 2C 3C C2 2C2 7C2+C

Find the value of C and also calculate the mean of this distribution.


Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day


Solve the following:

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

A highway safety group is interested in studying the speed (km/hrs) of a car at a check point.


The p.d.f. of r.v. of X is given by

f (x) = `k /sqrtx` , for 0 < x < 4 and = 0, otherwise. Determine k .

Determine c.d.f. of X and hence P (X ≤ 2) and P(X ≤ 1).


Determine whether each of the following is a probability distribution. Give reasons for your answer.

z 3 2 1 0 -1
P(z) 0.3 0.2 0.4. 0.05 0.05

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.


A coin is biased so that the head is 3 times as likely to occur as tail. Find the probability distribution of number of tails in two tosses.


There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?


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


10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?


Defects on plywood sheet occur at random with the average of one defect per 50 Sq.ft. Find the probability that such a sheet has no defect


Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:

  1. no defect
  2. at least one defect
    Use e−1 = 0.3678

State whether the following is True or False :

If r.v. X assumes the values 1, 2, 3, ……. 9 with equal probabilities, E(x) = 5.


Solve the following problem :

Following is the probability distribution of a r.v.X.

x – 3 – 2 –1 0 1 2 3
P(X = x) 0.05 0.1 0.15 0.20 0.25 0.15 0.1

Find the probability that X is non-negative


Solve the following problem :

Following is the probability distribution of a r.v.X.

x – 3 – 2 –1 0 1 2 3
P(X = x) 0.05 0.1 0.15 0.20 0.25 0.15 0.1

Find the probability that X is even.


Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as number greater than 4.


Solve the following problem :

Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.


Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows 3 heads


Solve the following problem :

The probability that a bomb will hit the target is 0.8. Find the probability that, out of 5 bombs, exactly 2 will miss the target.


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 :

A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Find the probability that the inspector finds at most one defective item in the 4 selected items.


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.


Find the mean and variance of the number randomly selected from 1 to 15


Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X.


A discrete random variable X has the probability distribution given as below:

X 0.5 1 1.5 2
P(X) k k2 2k2 k

Find 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 the value of k.


The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P(X = 1) = p and that E(X2) = E[X], find the value of p


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 ______.


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