Mathematics Chapter 17 Introduction To Mathematical Modelling

1500 families with 2 children were selected randomly, and the following data were recorded:

Compute the probability of a family, chosen at random, having

(0 2 girls (ii) 1 girl (iii) No girl.

Also check whether the sum of these probabilities is 1.

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.

An organisation selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below:

Suppose a family is chosen. Find the probability that the family chosen is
(i) earning  र 10000–13000 per month and owning exactly 2 vehicles.
(ii) earning र 16000 or more per month and owning exactly l vehicle.
(iii) earning less than र 7000 per month and does not own any vehicle.
(iv) earning र 13000–16000 per month and owning more than 2 vehicles.
(v) owning not more than I vehicle.

(i)  Find the probability that a student obtained less than 20% in the mathematics test.
(ii)  Find the probability that a student obtained marks 60 or above.

To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table:

Find the probability that a student chosen at random
(i)  likes statistics,
(ii)  does not like it.

Refer to Q.2, Exercise 14.2. What is the empirical probability that an engineer lives: (i) less than 7 km from her place of work? (ii) more than or equal to 7 km from her place of work? (iii) within  km from her place of work?

Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg):

4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00

Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

In Q.5, Exercise 14.2, you were asked to prepare a frequency distribution table, regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days. Using this table, find the probability of the concentration of sulphur dioxide in the interval 0.12 - 0.16 on any of these days

In Q.1, Exercise 14.2, you were asked to prepare a frequency distribution table regarding the blood groups of 30 students of a class. Use this table to determine the probability that a student of this class, selected at random, has blood group AB. 

A and B are the only two outcomes of an event. Probability P(A) = 0.72 then what will be the probability P(B) and why?

A survey of 500 families was conducted to know their opinion about a particular detergent powder. If 375 families liked the detergent powder and the remaining families disliked it, find the probability that a family chosen at random

(i)  likes the detergent powder
(ii)  does not like it.

1500 families with 2 children were released randomly and the following data was recorded:

If a family is chosen at random, find the probability that it has

(i)  at most one girl
(ii)  at least one girl 

Out of the past 250 consecutive days, its weather forecasts were correct 175 times.

(i)  What is the probability that on a given day it was correct?
(ii)  What is the probability that it was not correct on a given day?

A die is rolled 25 times and outcomes are recorded as under:

It is thrown one more time. Find the probability of getting

(a)  an even number
(b)  a multiple of 3
(c)  a prime number.

A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is

(i)  black
(ii)  not green.

Cards marked with numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is
(a)  a number less than 14
(b)  a number which is a perfect square
(c)  a prime number less than 20.

On a particular day, the number of vehicles passing through a crossing is given below:

A particular vehicle is chosen at random. What is the probability that it is not a four-wheeler?

30 plants were planted in each school out of 12 schools. After a month the number of plants that survived are given below:

What is the probability of survival of
(i)  more than 20 plants in a school?
(ii)  less than 10 plants in a school?
(iii) exactly 22 plants in a school?

The percentages of marks obtained by a student in examination are given below:

Find the probability that the student gets

(i)   a first class i.e. at least 60% marks
(ii)  a distinction i.e. 75% or above
(ii)  marks between 70% and 80%.

At a hospital, a doctor compiled the following data about 400 patients whom he could cure of hepatitis:

Another case of hepatitis is reported. What is the probability that this patient will be cured in

(i)   less than 2 months?
(ii)  1 month or more but not more than 3 months?

An insurance company selected 1600 drivers at random in a particular city to find a relationship between age and number of accidents. The data obtained are given in the following table:

Find the number of drivers
(a)  in the age of 25–40 years and has more than 2 accidents in the year.
(b)  the age is above 40 years and has accidents more than 1 but less than 3.

On a busy road, following data was observed about cars passing through it and number of occupants

Suppose another car passes by. Find the chance that it has
(i)  exactly 5 occupants
(ii)  more than 2 occupants (iii) less than 5 occupants.

The king, queen and jack of clubs are removed from a deck of 52 cards and then well shuffled. One card is selected at random from the remaining cards. Find the probability of getting

(a)  a heart
(b)  a king
(c)  the 10 of hearts.

A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.

Two coins are tossed simultaneously 500 times, and we get

Two heads: 105 times
One head: 275 times
No head: 120 times
Find the probability of occurrence of each of these events.

A dice is thrown 1000 times with the following frequencies for the outcomes 1, 2, 3, 4, 5 and 6 given in the following table:

Table

Find the probability of the happening of each outcome. 

On one page of a telephone directory, there were 200 telephone numbers. The frequency distribution of their unit place digit (for example, in the number 25828573, the unit place digit is 3) is given in the table below:

Table

Without looking at the page, the pencil is placed on one of these numbers, i.e., the number is chosen at random. What is the probability that the digit in its unit place is 6?

The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times:

(i)  What is the probability that on a given day it was correct?
(ii)  What is the probability that it was not correct on a given day? 

A tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of 1000 cases.  [CBSE 2012 (March)]

Table

If you buy a tyre of this company, what is the probability that:

(i) it will need to be replaced before it has covered 400 km?
(ii)  it will last more than 900 km?
(iii)  it will need to be replaced after it has covered somewhere between 400 km andb 1400 km?
(iv)  it will not need to be replaced at all?
(v)  it will need to be replaced?

The percentage of marks obtained by a student in the monthly unit tests are given below:
Table

Based on this data, find the probability that the student gets more than 70% marks in a unit test.

An insurance company selected 2000 drivers at random (i.e., without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table:

Table

Find the probabilities of the following events for a driver chosen at random from the city:

(i) being 18–29 years of age and having exactly 3 accidents in one year.
(ii)  being 30-50 years of age and having one or more accidents in a year.
(iii)  having no accidents in one year.

Consider the following frequency distribution table, which gives the weights of 38 students of a class:

(i)  Find the probability that the weight of a student in the class lies in the interval 46-50 kg.

(ii)  Give two events in this context, one having probability 0 and the other having probability 1:

Table

Fifty seeds were selected at random from each of 5 bags of seeds, and were kept under standardised conditions favourable to germination. After 20 days the number of seeds which had germinated in each collection were counted and recorded as follows:

Table

What is the probability of germination of
(i)   more than 40 seeds in a bag?
(ii)  49 seeds in a bag?
(iii)  more than 35 seeds in a bag?

Two coins are tossed simultaneously 1000 times with the following frequencies of different outcomes.

If these two coins are tossed again, find the probability of getting

(i)  at least 1 head
(ii)  at most 1 head

Two coins are tossed simultaneously 500 times,

and we get

Find the probability of occurrence of

(i)  two heads
(ii)  all tails.

These coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Compute the probability of getting

(i)  less than 2 heads
(ii)  3 heads.

A bag has 3 red and 7 black balls. One ball is taken out of the bag. Find the probability that it is a

(i)  red ball
(ii)  black ball. 

Cards marked with numbers 2, 3, 4.....61 are

placed in a box and mixed thoroughly. One card is drawn. Find the probability that card drawn in

(i)  an even number
(ii)  a square number.

1000 families with 2 children mere selected randomly, and the following data were recorded:

Compute the probability of a family, chosen at random having (0 2 girls

(ii)  1 girl

(iii)  No girl

Three coins are tossed simultaneously 250 times with the following frequencies of different outcomes:

If the three coins are simultaneously tossed again, find the probability of

(i)  3 heads
(ii)  less than 2 heads
(iii)  2 heads

In a box, there are 9 red, 8 white and 3 black balls. One ball is taken out of the bag. Find the probability that it is
(i)  white
(ii)  red or black
(iii)  not red. 

The following table represents the weight of 50 students of a school. A student is selected at random from the school. Find the probability of a student having weight of less than 60 kg.

 Given below is the frequency distribution of salary (in र) of 100 workers in a factory:

Answer the following questions. How many workers

(i)  have salary below र3000?
(ii)  have salary between र3000 and र5000?
(iii)  from र1000 to र5000?

A die was rolled 100 times and the number of times 6 came up was noted. If the experimental probability calculated from this information is 2/5 then how many times 6 came up? Justify your answer. 

In a survey of 500 families, number of vehicles owned per family were found to be as follows:

Find the probability that a family selected at random has

(i)  no vehicles
(ii)  2 or 3 vehicles
(iii)  more than 1 vehicle.

A die is thrown 1000 times with the following frequencies for the outcomes 1,2,3,4,5 and 6 as given below:

If the same die is thrown once more, find the probability of outcome 2, 4 and 6.

The ages of workers in a factory are given in the following table:

Find the probability that the age of worker selected at random is at least 25 years.

The minimum probablity of an event is 

• 0

• 1

• 

• - 1

The maximum probablity of an event is 

• 0

• 1

• 

• none of these

• 1

• 2

• 4

• 6

• 2

• 4

• 6

• 3

• 0

• 1

• - 1

• 2

• 0 ≤ P(E) ≤1
• 0 < P(E) < 1
• 0 ≤ P(E) < 1
• 0 < P(E) ≤ 1

A coin is tossed 100 times with the following frequencies:

Head : 75, Tail: 25

Find the probability of getting a head.

•  

• 
• 

• 1

A dice is thrown 100 times with the frequencies for the outcomes 1, 2, 3, 4, 5 and 6 as given in the following table:

Find the probability of outcome 1

Two coins are tossed simultaneously 1000 times, and we get
Two heads: 200 times
One head: 600 times
No head: 200 times
Find the probability of getting two heads.

Three coins one tossed simultaneously 600 times with the following frequencies of different outcomes:

The probability of getting 3 heads is

Three coins were tossed 30 times simultaneously. Each time the number of heads occurring was noted down as follows:

0 1 2 2 1 2
3 1 3 0 1 3
1 1 2 2 0 1
2 1 3 0 0 1
1 2 3 2 2 0

Find the probability of getting no head.

The following table shows the birth day of 30 students of class X:

Find the probability that a student was bom on Saturday.

The following table gives the distribution of IQ (intelligence quotient) of 60 pupils of class IX in a school:

Find the probability that a student of class IX selected at random has IQ in the interval 80-90.

The following table depicts the scores of 200 students in a test:

Find the probability that a student selected at random has his score in the interval 650-700.

Following are the marks of a group of 100 students in a test of reading ability:

Find the probability that a student selected at random gets marks in the interval 38–40.

The following distribution gives the times taken by 25 students to solve a problem:

Find the probability that the time taken by a student to solve the problem lies in the interval 35–40 or 45–50.

The following table presents the number of literate females in a town:

Find the probability that a literate female in from age group 10–15.

The distribution of weights (in kg) of 100 people is given below:

Find the probability that the weight of a people selected at random is in the class 45-50.

The blood groups of 30 students of Class VIII are recorded as follows:

A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O,
A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O.

Find the probability that a student of Class VIII selected at random has his blood group AB.

• 
• 
• 

• none of these

• 0.38
• 0.68
• 0.78
• 0.32

• 
• 

• 1

• 

• 
• 

• 1

• 0

• 1

• 0

• 

• 2

•  p ≤ 0 
•  0 < p ≤ 1
• 0 ≤ p ≤ 1 

• –1 ≤ p ≤ 1

   

• 0

• 1

• - 1

• 2

A coin is tossed 100 times with the following frequencies:  
Head — 45, Tail — 55 Then the probability of getting a head will be

• 0.45

• 0.55

• 0.40

• 0.50

• 
• 
• 

• 1

In a survey it was found that out of 364 persons, 91 persons like potato chips. If one person is selected at random, then the probability that he likes potato chips is:

• 0.75
• 0.50
• 0.25 
• 0.80

• 1

• 

• 2

• 

• 1

• 0

• 2

• 

• 15%
• 43%
• 57%
• 100%

When a coin is tossed 500 times, the following outcomes were recorded:
Head: 270 times, Tail: 230 times
If now a coin is tossed, the probability of getting a Head is

• 

• 

• 1

• 

A coin is tossed 500 times with the following frequencies:
Head: 235, Tail = 265
The coin is tossed once again, the probability of getting a Tail is

• 0.50

• 0.47
• 0.45
• 0.53

• less than 1
• greater than 1
• lies between 0 and 1
• 1

A dice is tossed 100 times. The frequency of occurrence of numbers 1,2,3,4,5,6 are given as under:
1 : 22, 2:28, 3: 12, 4: 14, 5: 11,6: 13
A die is tossed again. The probability of an odd prime number is

• 0.23
• 0.50
• 0.49 
• 0.73

A coin is tossed 1000 times with the following frequencies: 
Head: 460, Tail: 540
The coin is tossed again once. The probability of getting a tail is

• 15125
• 500
• 275
• 0.45

A coin is tossed 500 times with the following observations: Head: 245 times, Tail: 255 times The coin is tossed again. The probability of getting a Head is

• 0.502 
• 0.50
• 0.51  
• 0.49

• P(E) = 0
• P(E) = 1
• 0 ≥ P(E) > 1
• 0 ≤  P(E) ≤ 1

In a cricket match, a batsman hits a boundary 8 times out of the balls he plays. Find the balls played if the probability of not hitting a boundary is 80%:

• 10 
• 16
• 40 
• 64