problems on probability

 OBJECTIVE 

1. If there coins are tossed simultaneously, then the probability of getting at least two heads, is

(A) 1/4 (B) 3/8 (C) 1/2 (D) 1/4

2. A bag contains three green marbles four blue marbles, and two orange marbles. If marble is picked at random, then the probability that it is not a orange marble is

(A) 1/4  (B) 1/3 (C) 4/9  (D) 7/9

3. A number is selected from number 1 to 27. The probability that it is prime is

(A) 2/3  (B) 1/6 (C) 1/3 (D) 2/9

4. IF (P(E) = 0.05, then P (not E) =

(A) -0.05 (B) 0.5 (C) 0.9 (D) 0.95

5. A bulb is taken out at random from a box of 600 electric bulbs that contains 12 defective bulbs. Then the probability of a non-defective bulb is

(A) 0.02 (B) 0.98 (C) 0.50 (D) None

SUBJECTIVE 

1. To dice are thrown simultaneously. Find the probability of getting :

(i) An even number of the sum

(ii) The sum as a prime number

(iii) A total of at least 10

(iv) A multiple of 2 on one dice and a multiple of 3 on the other.

2. Find the probability that a leap year selected at random will contain 53 Tuesdays.

3. A bag contains 12 balls out of which x are white.(i) If one ball is drawn at random, what is the probability it will be a white ball ? (ii) If 6 more white balls are put in the box. The probability of drawing a white ball will be double than that is (i). Find x.

4. In a class, there are 18 girls and 16 boys. The class teacher wants to choose one pupil for class monitor. What she does, she writes the name of each pupil a card and puts them into a basket and mixes thoroughly. A child is asked to pick one card from the basket. What is the probability that the name written on the card is: (i) The name of a girl (ii) The name of boy ?

5. The probability of selecting a green marble at random from a jar that contains only green, white and yellow marbles is 1/4. The probability of selecting a white marble from the same jar is 1/3. If this jar contains 10 yellow marbles. What is the total number of marbles in the jar ?

6. A card is drawn at random from a well suffled desk of playing cards. Find the probability that the card drawn is (i) A card of spade or an ace (ii) A red king (iii) Neither a king nor a queen (iv) Either a king or a queen

7. There are 30 cards of same size in a bag on which number 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number of the selected card is not divisible by 3.

8. In figure points A,B,C and D are the centers of four circles that each have a radius of length on unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region ?

9. A bag contains 5 white balls, 6 red balls, 6 black balls and 8 green balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is (i) White (ii) Red or black (iii) Note green (iv) Neither white nor black  [CBSE - 2006]

10. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball. [CBSE - 2008]

11. Cards. marked with number 5 to 50, are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is (i) a prime number less than 10. (ii) a number which is a perfect square.

#jee #math #probability 

Probability

1.1 EXPERIMENT : The word experiment means an operation, which can produce well defined outcomes. 

The are two types of experiment : 

                   (i) Deterministic experiment 

                    (ii) Probabilistic or Random experiment 

(i) Deterministic Experiment : Those experiment which when repeated under identical conditions, produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

 (ii) Probabilistic or Random Experiment :- In an experiment, when repeated under identical conditions donot produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment. 

Sample space : The set of all possible out comes of a random experiment is called a sample space associated with it and is generally denoted by S. 

For example, When a dice is tossed then S = {1, 2, 3, 4, 5, 6}.

 Event : A subset of sample space associated with a random experiment is called an event. For example, In tossing a dive getting an even no is an event. 

Favorable Event : Let S be a sample space associated with a random experiment and A be event associated with the random experiment. The elementary events belonging to A are know as favorable events to the event A. 

For example, in throwing a pair of dive, A is defined by “Getting 8 as the sum”. Then following elementary events are as out comes : (2, 6), (3, 5), (4, 4) (5, 3) (6, 2). So, there are 5 elementary events favorable to event A. 

1.2 PROBABILITY : If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of happening or occurrence of event A is denoted by P(A)

 Thus,

    

P(A)     =   (Total number of favourable outcomes)/(Total number of possible outcomes)

                   =  m/n 

And 0 £ P(A) £ 1

 If, P(A) = 0, then A is called impossible event

If, P(A) = 1, then A is called sure event

                                P(A) + P (A") = 1

Where P(A) = probability of occurrence of A. 

            P (A') = probability of non - occurrence of A.

 

Ex.1 A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

Sol. Total number of balls in the bag = 5 + 4 + 7 = 16

\ Total number of elementary events =16

(i)                There are 7 white balls in the bag.

\ Favorable number of elementary events = 7

P(Getting a white ball ) = (Total No. of elementaryevents) /(Total No. favourable elementaryevents)

                    = 7 /16

 (ii) There are 4 balls that are neither red nor white 

 Favorable number of elementary events = 4 

Hence, P (Getting neither red not white ball) = 4/16 =1/4

 Ex.2 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting  [CBSE - 2007] (i) black face card                     (ii) a queen             (iii) a black card.

 Sol. After removing three face cards of spades (king, queen, jack) from a deck of 52 playing cards, there are 49 cards left in the pack. 

Out of these 49 cards one card can be chosen in 49 ways. 

Therefore, Total number of elementary events = 49 

(i) There are 6 black face cards out of which 3 face cards of spades are already removed. 

So, out of remaining 3 black face cards one black face card ban be chosen in 3 ways. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a black face card ) = 3/49

 (ii) There are 3 queens in the remaining 49 cards.

 So, out of these three queens, on queen can be chosen in 3 ways 

Therefore, Favorable number of elementary events = 3

 Hence P (Getting a queen) = 3/49

 (iii) There are 23 black cards in the remaining 49 cards, 

So, out to these 23 black card, one black card can be chosen in 23 ways 

Therefore, Favorable number of elementary events = 23 

Hence, P (Getting a black card) =  23/49

 Ex.3 A die is thrown, Find the probability of (i) prime number (ii) multiple of 2 or 3 (iii) a number greater than 3.

Sol.

In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. 

Therefore, the tome number of elementary events associated with the random experiment of throwing a die is 6.

 (i) Let A denote the event “Getting a prime no”. 

Clearly, event A occurs if any one of 2,3,5 comes as out come. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a prime no.) = 3/6=1/2

(ii) An multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as out comes.

Therefore, Favorable number of elementary events = 4 

Hence, P (Getting multiple of 2 or 3) = 4/6=2/3

 (iii) The event “Getting a number greater than 3” will occur, if we obtain one of number 4,5,6 as an out come. 

Therefore, Favorable number of out comes = 3

 Hence, required probability = 3/6=1/2

 Ex.4 Two unbiased coins are tossed simultaneously. Find the probability of getting (i) two heads (ii) at least one head (iii) at most one head. 

Sol. 

If two unbiased coins are tossed simultaneously, 

we obtain any one of the following as an out come : HH, HT, TH, TT 

Therefore, Total number of elementary events = 4

 (i) Two heads are obtained if elementary event HH occurs. 

Therefore, Favorable number of events = 1 Hence, P (Two heads) =  1/4

 (ii) At least one head is obtained if any one of the following elementary events happen : HH, HT, TH 

Therefore, favorable number of events = 3 

Hence P (At least one head) = 3/4

 (iii) If one of the elementary events HT, TH, TT occurs, than at most one head is obtained 

Therefore, favorable number of events = 3 

Hence, P (At most one head) = 3/3

Ex.5 A box contains 20 balls bearing numbers, 1,2,3,4…...20. A ball is drawn at random from the box. What is the probability that the number of the ball is (i) an odd number (ii) divisible by 2 or 3 (iii) prime number 

Sol. Here, total numbers are 20. 

 Total number of elementary events = 20 

(i) The number selected will be odd number, if it is elected from 1,3,5,7,9,11,13,15,17,19 

 Therefore, Favorable number of elementary events = 10 

Hence, P (An odd number ) = 10/20=1/2

 (ii) Number divisible by 2 or 3 are 2,3,4,6,8,9,10,12,14,15,16,18,20 

Therefore, Favorable number of elementary events = 13

 P (Number divisible by 2 or 3) =  13/20

 (iii) There are 8 prime number from 1 to 20 i.e., 2,3,5,7,11,13,17,19

Therefore, Favorable number of elementary events = 8 

P (prime number ) = 8/20=2/5

 Ex.6 A die is drop at random on the rectangular region as shown in figure. What is the probability that it will land inside the circle with diameter 1m ? 

comment answer of this question

New syllabus of class 9. Term wise. COURSE STRUCTURE CLASS –IX (2021-22)

COURSE STRUCTURE 

CLASS –IX (2021-22)

 FIRST TERM

UNIT- NUMBER SYSTEMS

1. NUMBER SYSTEM

Review of representation of natural numbers, integers, rational numbers on the number 

line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.

1. Examples of non-recurring/non-terminating decimals. Existence of non-rational 

numbers (irrational numbers) such as , √2,√3 and their representation on the number

2. Rationalization (with precise meaning) of real numbers of the type 1/(𝑎+𝑏√𝑥) and 1/(√𝑥+√√𝑦)

(and their combinations) where x and y are natural number and a and b are integers. 

3. Recall of laws of exponents with integral powers. Rational exponents with positive 

real bases (to be done by particular cases, allowing learner to arrive at the general 

laws.)

UNIT-ALGEBRA

2. LINEAR EQUATIONS IN TWO VARIABLES

Recall of linear equations in one variable.

 Introduction to the equation in two variables.

Focus on linear equations of the type ax+by+c=0. Explain that a linear equation in two

variables has infinitely many solutions and justify their being written as ordered pairs of real

numbers, plotting them and showing that they lie on a line. Graph of linear equations in two

variables. Examples, problems from real life with algebraic and graphical solutions being

done simultaneously

UNIT-COORDINATE GEOMETRY

3. COORDINATE GEOMETRY

The Cartesian plane, coordinates of a point, names and terms associated with the

coordinate plane, notations, plotting points in the plane.

UNIT-GEOMETRY

4. LINES AND ANGLES

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is

180˚ and the converse.

2. (Prove) If two lines intersect, vertically opposite angles are equal.

3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a

transversal intersects two parallel lines.

4. (Motivate) Lines which are parallel to a given line are parallel.

5. (Prove) The sum of the angles of a triangle is 180˚.

6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the

sum of the two interior opposite angles.

5. TRIANGLES

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one

triangle is equal to any two sides and the included angle of the other triangle (SAS

Congruence).

2. (Motivate) Two triangles are congruent if any two angles and the included side of one

triangle is equal to any two angles and the included side of the other triangle (ASA

Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three

sides of the other triangle (SSS Congruence).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle

are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS

Congruence)

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

7. (Motivate) The sides opposite to equal angles of a triangle are equal.

UNIT-MENSURATION

6. HERON’S FORMULA

Area of a triangle using Heron's formula (without proof)

UNIT-STATISTICS & PROBABILITY

7. STATISTICS

Introduction to Statistics: Collection of data, presentation of data — tabular form, ungrouped

/ grouped, bar graphs, histograms

SECOND TERM

UNIT-ALGEBRA

1. POLYNOMIALS

Definition of a polynomial in one variable, with examples and counter examples. Coefficients

of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant,

linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and

multiples. Zeros of a polynomial. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real

numbers, and of cubic polynomials using the Factor Theorem.

Recall of algebraic expressions and identities. Verification of identities

and their use in factorization of polynomials.

UNIT-GEOMETRY

2. QUADRILATERALS

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and

equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel

to the third side and in half of it and (motivate) its converse.

3. CIRCLES

Through examples, arrive at definition of circle and related concepts-radius, circumference,

diameter, chord, arc, secant, sector, segment, subtended angle.

1. (Prove) Equal chords of a circle subtend equal angles at the centre and (motivate) its

converse.

2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and

conversely, the line drawn through the centre of a circle to bisect a chord is

perpendicular to the chord.

3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the

centre (or their respective centres) and conversely.

4. (Motivate) The angle subtended by an arc at the centre is double the angle subtended

by it at any point on the remaining part of the circle.

5. (Motivate) Angles in the same segment of a circle are equal.

6. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is

180° and its converse.

4. CONSTRUCTIONS

1. Construction of bisectors of line segments and angles of measure 60˚, 90˚, 45˚ etc.,

equilateral triangles.

2. Construction of a triangle given its base, sum/difference of the other two sides and one

base angle.

UNIT-MENSURATION

5. SURFACE AREAS AND VOLUMES

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right

circular cylinders/cones.

UNIT-STATISTICS & PROBABILITY

6. PROBABILITY

History, Repeated experiments and observed frequency approach to probability. Focus is

on empirical probability. (A large amount of time to be devoted to group and to individual

activities to motivate the concept; the experiments to be drawn from real - life situations, and

from examples used in the chapter on statistics).