15. Probability
NCERT
15. Probability
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Important Note :
1. An event for an experiment is the collection of some outcomes of the experiment.
2. A trial is an action which results in one or several outcomes .
3. The empirical (or experimental) probability 
of an event 
is given by
4. (i) For one coin :
The sample space 
H , T 
(ii) For two coins :
The sample space HH , HT , TH , TT
(iii) For three coins :
The sample space HHH , HTT , HHT , HTH , THT , THH , TTH , TTT
(iv) For one dice :
The sample space 1 , 2 , 3 , 4 , 5 , 6
(v) For two dice :
The sample space (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1), (3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)
5. The playing card consists of 52 cards which are divided into 4 suits of 13 cards each :
(i) Spades (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2
(ii) Clubs (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2
(iii) Hearts (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2
(iv) Diamonds (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2
(v) Kings, queens and jacks are called face cards.
6. Prime number 1 to 100 are :
2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 .
7. The Probability of an event lies between 0 and 1 (0 and 1 inclusive).
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EXERCISE 15.1
1. In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays . Find the probability that she did not hit a boundary .
2. 1500 families with 2 children were selected randomly, and the following data were recorded :
|
Number of girls in a family |
2 |
1 |
0 |
|
Number of families |
475 |
814 |
211 |
Compute the probability of a family , choose at random, having
(i) 2 girls (ii) 1 girl (iii) No girl
Also check whether the sum of these probabilities is 1 .
3. In a particular section of Class IX , 40 students were asked about the month of their birth and the following graph was prepared for the data so obtained :
Graph photo page 247
Observe the bar graph given above and answer the following questions :
Find the probability that a student of the class was born in August .
4. Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes :
|
Outcome |
3 heads |
2 heads |
1 head |
No head |
|
Frequency |
23 |
72 |
77 |
28 |
If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up .
5. An organization 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 :
|
Monthly (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 59 82 |
0 2 1 25 88 |
Suppose a family is chosen . Find the probability that the family chosen is
(i) earning Rs 10000 – 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 vehicles .
(iv) earning Rs 13000 – 16000 per month and owning more than 2 vehicles .
(v) owning not more than 1 vehicle .
6. A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks .
|
Marks |
Number of students |
|
0 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – above |
7 10 10 20 20 15 8 |
|
Total |
90 |
(i) Find the probability that a student obtained less than 20% in the mathematics test .
(ii) Find the probability that a student obtained .
7. To know the opinion of the students about the subject statistics , a survey of 200 students was conducted . The data is recording in the following table .
|
Opinion |
Number of students |
|
Like Dislike |
135 65 |
Find the probability that a student chosen at random (i) likes statistics (ii) does not like it .
8. The distance (in km) of 40 engineers from their residence to their place of work were found as follows :
5 3 10 20 25 11 13 7 12 31 19 10 12 17 18 11 32 17 16 2 7 9 7 8 3 5 12 15 18 3 12 14 2 9 6 15 15 7 6 12
Construct a grouped frequency distribution table with class size 5 for the data given above taking the first interval as 0 – 5 (5 not included) . 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 ?
9. Activity : Note the frequency of two-wheelers, three-wheeler and four-wheelers going past during a time interval , in front of your school gate . Find the probability that any one vehicle out of the total vehicles you have observed is a two-wheeler .
10. Activity : Ask all the students in your class to write a 3-digit number . Choose any student from the room at random . What is the probability that the number written by her/him is divisible by 3 ? Remember that a number is divisible by 3 , if the sum of its digits is divisible by 3 .
11. 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 .
12. A study was conducted to find out the concentration of sulphur dioxide in the air in parts per million (ppm)of a certain city .The data obtained for 30 days is as follows: 0.03 0.08 0.08 0.09 0.04 0.17 0.16 0.05 0.02 0.06 0.18 0.20 0.11 0.08 0.12 0.13 0.22 0.07 0.08 0.01 0.10 0.06 0.09 0.18 0.11 0.07 0.05 0.07 0.01 0.04
Using this table , find the probability of the concentration of sulphur dioxide in the interval 0.12 – 0.16 on any of these days .
13. The blood group 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.
Use this table to determine the probability that a student of this class, selected at random, has blood group AB .
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