Exercise 25.1
Page-25.13Question 1:
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
Answer 1:
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted by
and is given by
Therefore, we have
Question 2:
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
Answer 2:
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is 115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
Therefore, we have
Question 3:
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
| Outcome: | No head | One head | Two heads | Three heads |
| Frequency: | 14 | 38 | 36 | 12 |
If the three coins are simultaneously tossed again, compute the probability of:
(i) 2 heads coming up.
(ii) 3 heads coming up.
(iii) at least one head coming up.
(iv) getting more heads than tails.
(v) getting more tails than heads.
Answer 3:
The total number of trials is 100.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A be the event of getting two heads.
The number of times A happens is 36.
Therefore, we have
(ii) Let B be the event of getting three heads
The number of times B happens is 12.
Therefore, we have
(iii) Let C be the event of getting at least one head.
The number of times C happens is.
Therefore, we have
(iv) Let D be the event of getting more heads than tails.
The number of times D happens is.
Therefore, we have
(v) Let E be the event of getting more tails than heads.
The number of times E happens is.
Therefore, we have
Question 4:
1500 families with 2 children were selected randomly and the following data were recorded:
| Number of girls in a family | 0 | 1 | 2 |
| Number of families | 211 | 814 | 475 |
If a family is chosen at random, compute the probability that it has:
(i) No girl
(ii) 1 girl
(iii) 2 girls
(iv) at most one girl
(v) more girls than boys
Answer 4:
The total number of trials is 1500.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A be the event of having no girl.
The number of times A happens is 211.
Therefore, we have
(ii) Let B be the event of having one girl.
The number of times B happens is 814.
Therefore, we have
(iii) Let C be the event of having two girls.
The number of times C happens is 475.
Therefore, we have
(iv) Let D be the event of having at most one girl.
The number of times D happens is.
Therefore, we have
(v) Let E be the event of having more girls than boys.
The number of times E happens is 475.
Therefore, we have
Question 5:
In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.
(i) he hits boundary
(ii) he does not hit a boundary.
Answer 5:
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A be the event of hitting boundary.
The number of times A happens is 6.
Therefore, we have
(ii) Let B be the event of does not hitting boundary.
The number of times B happens is.
Therefore, we have
Question 6:
The percentage of marks obtained by a student in monthly unit tests are given below:
| Unit test: | I | II | III | IV | V |
| Percentage of marks obtained: | 69 | 71 | 73 | 68 | 76 |
Find the probability that the student gets:
(i) more than 70% marks
(ii) less than 70% marks
(iii) a distinction
Answer 6:
The total number of trials is 5.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A be the event of getting more than 70% marks.
The number of times A happens is 3.
Therefore, we have
(ii) Let B be the event of getting less than 70% marks.
The number of times B happens is 2.
Therefore, we have
(iii) Let C be the event of getting a distinction.
The number of times C happens is 1.
Therefore, we have
Question 7:
To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following table:
| Opinion: | Like | Dislike |
| Number of students: | 135 | 65 |
Find the probability that a student chosen at random (i) likes Mathematics (ii) does not like it.
Answer 7:
The total number of trials is 200.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted by
and is given by
(i) Let A be the event of liking mathematics.
The number of times A happens is 135.
Therefore, we have
(ii) Let B be the event of disliking mathematics.
The number of times B happens is 65.
Therefore, we have
Question 8:
The blood groups of 30 students of class IX 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 |
A student is selected at random from the class from blood donation, Fin the probability that the blood group of the student chosen is:
(i) A
(ii) B
(iii) AB
(iv) O
Answer 8:
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A1 be the event that the blood group of a chosen student is A.
The number of times A1 happens is 9.
Therefore, we have
(iii) Let A2 be the event that the blood group of a chosen student is B.
The number of times A2 happens is 6.
Therefore, we have
(iii) Let A3 be the event that the blood group of a chosen student is AB.
The number of times A3 happens is 3.
Therefore, we have
(iv) Let A4 be the event that the blood group of a chosen student is O.
The number of times A4 happens is 12.
Therefore, we have
Question 9:
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.
Answer 9:
The total number of trials is 11.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
Let A1 be the event that the actual weight of a chosen bag contain more than 5 Kg of flour.
The number of times A1 happens is 7.
Therefore, we have
Question 10:
Following table shows the birth month of 40 students of class IX.
| Jan. | Feb | March | April | May | June | July | Aug. | Sept. | Oct. | Nov. | Dec. |
| 3 | 4 | 2 | 2 | 5 | 1 | 2 | 5 | 3 | 4 | 4 | 4 |
Find the probability that a student was born in August.
Answer 10:
The total number of trials is 40.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
Let A1 be the event that the birth month of a chosen student is august.
The number of times A1 happens is 5.
Therefore, we have
Question 11:
Given below is the frequency distribution table regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days.
| Conc. of SO2 | 0.00-0.04 | 0.04-0.08 | 0.08-0.12 | 0.12-0.16 | 0.16-0.20 | 0.20-0.24 |
| No. days: | 4 | 8 | 9 | 2 | 4 | 3 |
Find the probability of concentration of sulphur dioxide in the interval
0.12-0.16 on any of these days.
Answer 11:
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
Let A1 be the event that the concentration of sulphur dioxide in a day is 0.12-0.16 parts per million.
The number of times A1 happens is 2.
Therefore, we have
Question 12:
A company selected 2400 families at random and survey them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed in the table below:
| Monthly income: (in Rs) |
Vehicles per family | |||
| 0 | 1 | 2 | Above 2 | |
| Less than 7000 7000-10000 10000-13000 13000-16000 16000 or more |
10 0 1 2 1 |
160 305 535 469 579 |
25 27 29 29 82 |
0 2 1 25 88 |
If a family is chosen, find the probability that family is:
(i) earning Rs10000-13000 per month and owning exactly 2 vehicles.
(ii) earning Rs 16000 or more per month and owning exactly 1 vehicle.
(iii) earning less than Rs 7000 per month and does not own any vehicle.
(iv) earning Rs 13000-16000 per month and owning more than 2 vehicle.
(v) owning not more than 1 vehicle
(vi) owning at least one vehicle.
Answer 12:
The total number of trials is 2400.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A1 be the event that a chosen family earns Rs 10000-13000 per month and owns exactly 2 vehicles.
The number of times A1 happens is 29.
Therefore, we have
(ii) Let A2 be the event that a chosen family earns Rs 16000 or more per month and owns exactly 1 vehicle.
The number of times A2 happens is 579.
Therefore, we have
(iii) Let A3 be the event that a chosen family earns less than Rs 7000 per month and does not owns any vehicles.
The number of times A3 happens is 10.
Therefore, we have
(iv) Let A4 be the event that a chosen family earns Rs 13000-16000 per month and owns more than 2 vehicles.
The number of times A4 happens is 25.
Therefore, we have
(v) Let A5 be the event that a chosen family owns not more than 1 vehicle (may be 0 or 1). In this case the number of vehicles is independent of the income of the family.
The number of times A5 happens is
.
Therefore, we have
(vi) Let A6 be the event that a chosen family owns atleast 1 vehicle (may be 1 or 2 or above 2). In this case the number of vehicles is independent of the income of the family.
The number of times A6 happens is
.
Therefore, we have
Question 13:
The following table gives the life time of 400 neon lamps:
| Life time (in hours) |
300-400 | 400-500 | 500-600 | 600-700 | 700-800 | 800-900 | 900-1000 |
| Number of lamps: | 14 | 56 | 60 | 86 | 74 | 62 | 48 |
A bulb is selected of random, Find the probability that the the life time of the selected bulb is:
(i) less than 400
(ii) between 300 to 800 hours
(iii) at least 700 hours.
Answer 13:
The total number of trials is 400.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A1 be the event that the lifetime of a chosen bulb is less than 400 hours.
The number of times A1 happens is 14.
Therefore, we have
(ii) Let A2 be the event that the lifetime of a chosen bulb is in between 300 to 800 hours.
The number of times A2 happens is
.
Therefore, we have
.
(iii) Let A3 be the event that the lifetime of a chosen bulb is atleast 700 hours.
The number of times A3 happens is
.
Therefore, we have
Question 14:
Given below is the frequency distribution of wages (in Rs) of 30 workers in a certain factory:
| Wages (in Rs) | 110-130 | 130-150 | 150-170 | 170-190 | 190-210 | 210-230 | 230-250 |
| No. of workers | 3 | 4 | 5 | 6 | 5 | 4 | 3 |
A worker is selected at random. Find the probability that his wages are:
(i) less than Rs 150
(ii) at least Rs 210
(iii) more than or equal to 150 but less than Rs 210.
Answer 14:
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by
(i) Let A1 be the event that the wages of a worker is less than Rs 150.
The number of times A1 happens is
.
Therefore, we have
.
(ii) Let A2 be the event that the wages of a worker is atleast Rs 210.
The number of times A2 happens is
.
Therefore, we have
.
(iii) Let A3 be the event that the wages of a worker is more than or equal to Rs 150 but less than Rs 210.
The number of times A3 happens is
.
Therefore, we have
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