RD Chapter 25 Probability Ex 25.1 Solutions
Question - 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 by and 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 by and 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
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 by and 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
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 by and 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
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 by and 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
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
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 by and 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
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