NCERT Exemplar Solutions
Class 10 - MathematicsCHAPTER 13: Statistics and Probability
Class 10 - Mathematics
Exercise 13.1
Question. 1
In the formula \(\bar{x}=a+\dfrac{\sum f_i d_i}{\sum f_i}\), for finding the mean of grouped data, \(d_i\)'s are deviations from \(a\) of
lower limits of the classes
upper limits of the classes
mid-points (class marks) of the classes
frequencies of the class marks
Open
Question. 2
While computing mean of grouped data, we assume that the frequencies are
evenly distributed over all the classes
centred at the class marks of the classes
centred at the upper limits of the classes
centred at the lower limits of the classes
Open
Question. 3
If \(x_i\) are the class marks and \(f_i\) the corresponding frequencies with mean \(\bar{x}\), then \(\sum f_i(x_i-\bar{x})\) equals
0
-1
1
2
Open
Question. 4
In the formula \(\bar{x}=a+h\,\dfrac{\sum f_i u_i}{\sum f_i}\) for mean of grouped data, \(u_i=\)
\(\dfrac{x_i+a}{h}\)
\(h(x_i-a)\)
\(\dfrac{x_i-a}{h}\)
\(\dfrac{a-x_i}{h}\)
Open
Question. 5
The abscissa of the intersection point of the less-than and more-than cumulative frequency curves of a grouped data gives its
mean
median
mode
all the three above
Open
Question. 6
For the following distribution:
| Class | 0–5 | 5–10 | 10–15 | 15–20 | 20–25 |
|---|---|---|---|---|---|
| Frequency | 10 | 15 | 12 | 20 | 9 |
the sum of lower limits of the median class and modal class is
15
25
30
35
Open
Question. 7
Consider the distribution:
| Class | 0–5 | 6–11 | 12–17 | 18–23 | 24–29 |
|---|---|---|---|---|---|
| Frequency | 13 | 10 | 15 | 8 | 11 |
The upper limit of the median class is
17
17.5
18
18.5
Open
Question. 8
For the following distribution:
| Marks | Below 10 | Below 20 | Below 30 | Below 40 | Below 50 | Below 60 |
|---|---|---|---|---|---|---|
| No. of students | 3 | 12 | 27 | 57 | 75 | 80 |
The modal class is
10–20
20–30
30–40
50–60
Open
Question. 9
Consider the data:
| Class | 65–85 | 85–105 | 105–125 | 125–145 | 145–165 | 165–185 | 185–205 |
|---|---|---|---|---|---|---|---|
| Frequency | 4 | 5 | 13 | 20 | 14 | 7 | 4 |
The difference of the upper limit of the median class and the lower limit of the modal class is
0
19
20
38
Open
Question. 10
Times (s) taken by 150 athletes in a 110 m hurdle race:
| Class | 13.8–14 | 14–14.2 | 14.2–14.4 | 14.4–14.6 | 14.6–14.8 | 14.8–15 |
|---|---|---|---|---|---|---|
| Frequency | 2 | 4 | 5 | 71 | 48 | 20 |
The number who finished in less than 14.6 s is
11
71
82
130
Open
Question. 11
Consider:
| Marks obtained | ≥0 | ≥10 | ≥20 | ≥30 | ≥40 | ≥50 |
|---|---|---|---|---|---|---|
| No. of students | 63 | 58 | 55 | 51 | 48 | 42 |
The frequency of the class 30–40 is
3
4
48
51
Open
Question. 12
If an event cannot occur, its probability is
1
\(\dfrac{3}{4}\)
\(\dfrac{1}{2}\)
0
Open
Question. 13
Which of the following cannot be the probability of an event?
\(\dfrac{1}{3}\)
0.1
3%
\(\dfrac{17}{16}\)
Open
Question. 14
An event is very unlikely to happen. Its probability is closest to
0.0001
0.001
0.01
0.1
Open
Question. 15
If the probability of an event is \(p\), the probability of its complementary event is
\(p-1\)
\(p\)
\(1-p\)
\(1-\dfrac{1}{p}\)
Open
Question. 16
The probability expressed as a percentage of a particular occurrence can never be
less than 100
less than 0
greater than 1
anything but a whole number
Open
Question. 17
If \(P(A)\) denotes the probability of an event A, then
\(P(A)<0\)
\(P(A)>1\)
\(0\le P(A)\le 1\)
\(-1\le P(A)\le 1\)
Open
Question. 18
A card is selected from a deck of 52 cards. The probability that it is a red face card is
\(\dfrac{3}{26}\)
\(\dfrac{3}{13}\)
\(\dfrac{2}{13}\)
\(\dfrac{1}{2}\)
Open
Question. 19
The probability that a non-leap year selected at random will contain 53 Sundays is
\(\dfrac{1}{7}\)
\(\dfrac{2}{7}\)
\(\dfrac{3}{7}\)
\(\dfrac{5}{7}\)
Open
Question. 20
When a die is thrown, the probability of getting an odd number less than 3 is
\(\dfrac{1}{6}\)
\(\dfrac{1}{3}\)
\(\dfrac{1}{2}\)
0
Open
Question. 21
A card is drawn from a deck of 52 cards. Event \(E\): “card is not an ace of hearts”. The number of outcomes favourable to \(E\) is
4
13
48
51
Open
Question. 22
The probability of getting a bad egg in a lot of 400 is 0.035. The number of bad eggs in the lot is
7
14
21
28
Open
Question. 23
A girl finds the probability of winning the first prize in a lottery to be 0.08. If 6000 tickets are sold, how many tickets has she bought?
40
240
480
750
Open
Question. 24
One ticket is drawn at random from a bag of tickets numbered 1 to 40. The probability that the selected ticket is a multiple of 5 is
\(\dfrac{1}{5}\)
\(\dfrac{3}{5}\)
\(\dfrac{4}{5}\)
\(\dfrac{1}{3}\)
Open
Question. 25
Someone is asked to take a number from 1 to 100. The probability that it is a prime is
\(\dfrac{1}{5}\)
\(\dfrac{6}{25}\)
\(\dfrac{1}{4}\)
\(\dfrac{13}{50}\)
Open
Question. 26
A class has 23 students: 4 from house A, 8 from B, 5 from C, 2 from D and the rest from E. One student is selected at random to be the monitor. The probability that the selected student is not from A, B and C is
\(\dfrac{4}{23}\)
\(\dfrac{6}{23}\)
\(\dfrac{8}{23}\)
\(\dfrac{17}{23}\)
Open
Exercise 13.2
Question. 1
The median of an ungrouped data and the median calculated when the same data is grouped are always the same. Do you think that this is a correct statement? Give reason.
Answer:
Open
Question. 2
In calculating the mean of grouped data, grouped in classes of equal width, we may use the formula
\[ \bar{x} = a + \dfrac{f_i d_i}{f_i} \]
where a is the assumed mean. a must be one of the mid–points of the classes. Is this correct? Justify your answer.
Answer:
Open
Question. 3
Is it true to say that the mean, mode and median of grouped data will always be different? Justify your answer.
Answer:
Open
Question. 4
Will the median class and modal class of grouped data always be different? Justify your answer.
Answer:
Open
Question. 5
In a family having three children, there may be no girl, one girl, two girls or three girls. So, the probability of each is \(\dfrac{1}{4}\). Is this correct? Justify your answer.
Answer:
Open
Question. 6
A game consists of spinning an arrow which comes to rest pointing at one of the regions (1, 2 or 3) 
. Are the outcomes 1, 2 and 3 equally likely to occur? Give reasons.
Answer:
Open
Question. 7
Apoorv throws two dice once and computes the product of the numbers appearing on the dice. Peehu throws one die and squares the number that appears on it. Who has the better chance of getting the number 36? Why?
Answer:
Open
Question. 8
When we toss a coin, there are two possible outcomes – Head or Tail. Therefore, the probability of each outcome is \(\dfrac{1}{2}\). Justify your answer.
Answer:
Open
Question. 9
A student says that if you throw a die, it will show up 1 or not 1. Therefore, the probability of getting 1 and the probability of getting ‘not 1’ each is equal to \(\dfrac{1}{2}\). Is this correct? Give reasons.
Answer:
Open
Question. 10
I toss three coins together. The possible outcomes are no heads, 1 head, 2 heads and 3 heads. So, I say that probability of no heads is \(\dfrac{1}{4}\). What is wrong with this conclusion?
Answer:
Open
Question. 11
If you toss a coin 6 times and it comes down heads on each occasion. Can you say that the probability of getting a head is 1? Give reasons.
Answer:
Open
Question. 12
Sushma tosses a coin 3 times and gets tail each time. Do you think that the outcome of next toss will be a tail? Give reasons.
Answer:
Open
Question. 13
If I toss a coin 3 times and get head each time, should I expect a tail to have a higher chance in the 4th toss? Give reason in support of your answer.
Answer:
Open
Question. 14
A bag contains slips numbered from 1 to 100. If Fatima chooses a slip at random from the bag, it will either be an odd number or an even number. Since this situation has only two possible outcomes, so, the probability of each is \(\dfrac{1}{2}\). Justify.
Answer:
Open
Exercise 13.3
Question. 1
1. Find the mean of the distribution :
| Class | 1–3 | 3–5 | 5–7 | 7–10 |
|---|---|---|---|---|
| Frequency | 9 | 22 | 27 | 17 |
Answer:
5.5
Open
Question. 2
2. Calculate the mean of the scores of 20 students in a mathematics test :
| Marks | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|
| Number of students | 2 | 4 | 7 | 6 | 1 |
Answer:
35
Open
Question. 3
3. Calculate the mean of the following data :
| Class | 4–7 | 8–11 | 12–15 | 16–19 |
|---|---|---|---|---|
| Frequency | 5 | 4 | 9 | 10 |
Answer:
\(\displaystyle 12.93\) (approx.)
Open
Question. 4
4. Pages written by Sarika in 30 days:
| Pages/day | 16–18 | 19–21 | 22–24 | 25–27 | 28–30 |
|---|---|---|---|---|---|
| Number of days | 1 | 3 | 4 | 9 | 13 |
Find the mean number of pages per day.
Answer:
26 pages/day
Open
Question. 5
5. Daily income (Rs) of 50 employees:
| Income (Rs) | 1–200 | 201–400 | 401–600 | 601–800 |
|---|---|---|---|---|
| No. of employees | 14 | 15 | 14 | 7 |
Find the mean daily income.
Answer:
Rs 356.5
Open
Question. 6
6. An aircraft has 120 seats. Over 100 flights, the seats occupied were:
| Seats | 100–104 | 104–108 | 108–112 | 112–116 | 116–120 |
|---|---|---|---|---|---|
| Frequency | 15 | 20 | 32 | 18 | 15 |
Determine the mean number of seats occupied.
Answer:
109.92 seats
Open
Question. 7
7. Weights (kg) of 50 wrestlers:
| Weight (kg) | 100–110 | 110–120 | 120–130 | 130–140 | 140–150 |
|---|---|---|---|---|---|
| No. of wrestlers | 4 | 14 | 21 | 8 | 3 |
Find the mean weight.
Answer:
123.4 kg
Open
Question. 8
8. Mileage (km/l) of 50 cars:
| Mileage | 10–12 | 12–14 | 14–16 | 16–18 |
|---|---|---|---|---|
| No. of cars | 7 | 12 | 18 | 13 |
Find the mean mileage. The manufacturer claimed the model gave 16 km/litre. Do you agree?
Answer:
14.48 km/l (Claim of 16 km/l is not supported.)
Open
Question. 9
9. Distribution of weights (kg) of 40 persons:
| Weight (kg) | 40–45 | 45–50 | 50–55 | 55–60 | 60–65 | 65–70 | 70–75 | 75–80 |
|---|---|---|---|---|---|---|---|---|
| No. of persons | 4 | 4 | 13 | 5 | 6 | 5 | 2 | 1 |
Construct the less-than type cumulative frequency table.
Answer:
| Less than | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 |
|---|---|---|---|---|---|---|---|---|
| Cumulative frequency | 4 | 8 | 21 | 26 | 32 | 37 | 39 | 40 |
Open
Question. 10
10. Cumulative frequency (less-than) of marks of 800 students:
| Marks | Below 10 | Below 20 | Below 30 | Below 40 | Below 50 | Below 60 | Below 70 | Below 80 | Below 90 | Below 100 |
|---|---|---|---|---|---|---|---|---|---|---|
| No. of students | 10 | 50 | 130 | 270 | 440 | 570 | 670 | 740 | 780 | 800 |
Construct the (ordinary) frequency distribution table.
Answer:
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 | 90–100 |
|---|---|---|---|---|---|---|---|---|---|---|
| Frequency | 10 | 40 | 80 | 140 | 170 | 130 | 100 | 70 | 40 | 20 |
Open
Question. 11
11. From the following “more than or equal to” data, form the frequency distribution:
| Marks (out of 90) | ≥80 | ≥70 | ≥60 | ≥50 | ≥40 | ≥30 | ≥20 | ≥10 | ≥0 |
|---|---|---|---|---|---|---|---|---|---|
| No. of candidates | 4 | 6 | 11 | 17 | 23 | 27 | 30 | 32 | 34 |
Answer:
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 |
|---|---|---|---|---|---|---|---|---|---|
| Frequency | 2 | 2 | 3 | 4 | 6 | 6 | 5 | 2 | 4 |
Open
Question. 12
12. Fill the unknown entries \(a,b,c,d,e,f\) in the following cumulative table:
| Height (cm) | Frequency | Cumulative frequency |
|---|---|---|
| 150–155 | 12 | a |
| 155–160 | b | 25 |
| 160–165 | 10 | c |
| 165–170 | d | 43 |
| 170–175 | e | 48 |
| 175–180 | 2 | f |
| Total | 50 |
Answer:
a=12, b=13, c=35, d=8, e=5, f=50
Open
Question. 13
13. Ages (years) of 300 patients on a day:
| Age | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 |
|---|---|---|---|---|---|---|
| No. of patients | 60 | 42 | 55 | 70 | 53 | 20 |
Form (i) Less-than type and (ii) More-than type cumulative frequency distributions.
Answer:
(i) Less-than type
| Less than | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|
| CF | 60 | 102 | 157 | 227 | 280 | 300 |
(ii) More-than type
| More than or equal to | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|---|
| CF | 300 | 240 | 198 | 143 | 73 | 20 | 0 |
Open
Question. 14
14. Given cumulative (less-than) marks of 50 students:
| Marks | Below 20 | Below 40 | Below 60 | Below 80 | Below 100 |
|---|---|---|---|---|---|
| No. of students | 17 | 22 | 29 | 37 | 50 |
Form the ordinary frequency distribution.
Answer:
| Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Frequency | 17 | 5 | 7 | 8 | 13 |
Open
Question. 15
15. Weekly income of 600 families:
| Income (Rs) | 0–1000 | 1000–2000 | 2000–3000 | 3000–4000 | 4000–5000 | 5000–6000 |
|---|---|---|---|---|---|---|
| No. of families | 250 | 190 | 100 | 40 | 15 | 5 |
Compute the median income.
Answer:
≈ Rs 1263.16
Open
Question. 16
16. Maximum bowling speeds (km/h) of 33 players:
| Speed | 85–100 | 100–115 | 115–130 | 130–145 |
|---|---|---|---|---|
| No. of players | 11 | 9 | 8 | 5 |
Calculate the median speed.
Answer:
≈ 109.17 km/h
Open
Question. 17
17. Monthly income of 100 families:
| Income (Rs) | 0–5000 | 5000–10000 | 10000–15000 | 15000–20000 | 20000–25000 | 25000–30000 | 30000–35000 | 35000–40000 |
|---|---|---|---|---|---|---|---|---|
| No. of families | 8 | 26 | 41 | 16 | 3 | 3 | 2 | 1 |
Calculate the modal income.
Answer:
≈ Rs 11,875
Open
Question. 18
18. Weights of 70 coffee packets:
| Weight (g) | 200–201 | 201–202 | 202–203 | 203–204 | 204–205 | 205–206 |
|---|---|---|---|---|---|---|
| No. of packets | 12 | 26 | 20 | 9 | 2 | 1 |
Determine the modal weight.
Answer:
≈ 201.7 g
Open
Question. 19
Two dice are thrown. Find the probability of getting (i) the same number on both, (ii) different numbers.
Answer:
(i) \(\dfrac{1}{6}\); (ii) \(\dfrac{5}{6}\)
Open
Question. 20
Two dice are thrown. Probability that the sum is (i) 7 (ii) a prime number (iii) 1?
Answer:
(i) \(\dfrac{1}{6}\), (ii) \(\dfrac{5}{12}\), (iii) \(0\)
Open
Question. 21
Two dice are thrown. Probability that the product is (i) 6 (ii) 12 (iii) 7?
Answer:
(i) \(\dfrac{1}{9}\), (ii) \(\dfrac{1}{9}\), (iii) \(0\)
Open
Question. 22
Two dice are thrown and the product of the numbers is noted. Probability that the product is less than 9?
Answer:
\(\dfrac{4}{9}\)
Open
Question. 23
Die I has faces 1–6. Die II has faces 1,1,2,2,3,3. They are thrown; find probabilities of sums 2 to 9 (separately).
Answer:
| Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| Probability | \(\dfrac{1}{18}\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{18}\) |
Open
Question. 24
A coin is tossed two times. Probability of getting at most one head?
Answer:
\(\dfrac{3}{4}\)
Open
Question. 25
A coin is tossed 3 times. List outcomes and find probability of (i) all heads (ii) at least two heads.
Answer:
(i) \(\dfrac{1}{8}\), (ii) \(\dfrac{1}{2}\)
Open
Question. 26
Two dice are thrown. Probability that the absolute difference of the numbers is 2?
Answer:
\(\dfrac{2}{9}\)
Open
Question. 27
A bag has 10 red, 5 blue, 7 green balls. Probability that a ball drawn is (i) red (ii) green (iii) not blue?
Answer:
(i) \(\dfrac{5}{11}\), (ii) \(\dfrac{7}{22}\), (iii) \(\dfrac{17}{22}\)
Open
Question. 28
From a deck, remove K, Q, J of clubs; draw one card from remaining. Probability that card is (i) a heart (ii) a king?
Answer:
(i) \(\dfrac{13}{49}\), (ii) \(\dfrac{3}{49}\)
Open
Question. 29
(Ref. Q28) Probability that the card is (i) a club (ii) 10 of hearts?
Answer:
(i) \(\dfrac{10}{49}\), (ii) \(\dfrac{1}{49}\)
Open
Question. 30
Remove all J, Q, K from a 52-card deck. With Ace valued 1, find probability that a drawn card has value (i) 7 (ii) >7 (iii) <7.
Answer:
(i) \(\dfrac{1}{10}\), (ii) \(\dfrac{3}{10}\), (iii) \(\dfrac{3}{5}\)
Open
Question. 31
An integer is chosen between 0 and 100 (inclusive). Probability it is (i) divisible by 7 (ii) not divisible by 7?
Answer:
(i) \(\dfrac{15}{101}\), (ii) \(\dfrac{86}{101}\)
Open
Question. 32
Cards numbered 2 to 101 are in a box (100 cards). Probability that the card has (i) an even number (ii) a square number?
Answer:
(i) \(\dfrac{1}{2}\), (ii) \(\dfrac{9}{100}\)
Open
Question. 33
A letter of the English alphabet is chosen at random. Probability it is a consonant?
Answer:
\(\dfrac{21}{26}\)
Open
Question. 34
1000 sealed envelopes: 10 contain Rs 100, 100 contain Rs 50, 200 contain Rs 10, rest contain no cash. If one is picked at random, probability it contains no cash prize?
Answer:
\(\dfrac{69}{100}\)
Open
Question. 35
Box A: 25 slips (19 marked Re 1, 6 marked Rs 5). Box B: 50 slips (45 marked Re 1, 5 marked Rs 13). Slips are mixed and one slip is drawn. Probability it is marked other than Re 1?
Answer:
\(\dfrac{11}{75}\)
Open
Question. 36
A carton of 24 bulbs has 6 defective. One bulb is drawn.
(i) Probability it is not defective. (ii) If the first drawn bulb is defective and not replaced, find the probability that a second bulb drawn is defective.
Answer:
(i) \(\dfrac{3}{4}\); (ii) \(\dfrac{5}{23}\)
Open
Question. 37
A child’s game has 8 triangles (3 blue, 5 red) and 10 squares (6 blue, 4 red). One piece is lost at random. Find the probability it is (i) a triangle (ii) a square (iii) a blue square (iv) a red triangle.
Answer:
(i) \(\dfrac{4}{9}\), (ii) \(\dfrac{5}{9}\), (iii) \(\dfrac{1}{3}\), (iv) \(\dfrac{5}{18}\)
Open
Question. 38
A game: Toss a coin 3 times. If 1 or 2 heads appear, Sweta gets her entry fee back; if 3 heads appear, she gets double back; otherwise she loses. Find probabilities that she (i) loses (ii) gets double (iii) just gets entry fee back.
Answer:
(i) \(\dfrac{1}{8}\), (ii) \(\dfrac{1}{8}\), (iii) \(\dfrac{3}{4}\)
Open
Question. 39
A die has faces 0,1,1,1,6,6. Two such dice are thrown and total score recorded. (i) How many different totals possible? (ii) Probability of a total of 7?
Answer:
(i) 6 totals (0,1,2,6,7,12); (ii) \(\dfrac{1}{3}\)
Open
Question. 40
Lot of 48 mobiles: 42 good, 3 minor defects, 3 major defects. Varnika buys only good; trader sells only if no major defect. A phone is selected at random. Probability it is (i) acceptable to Varnika (ii) acceptable to trader?
Answer:
(i) \(\dfrac{7}{8}\) ; (ii) \(\dfrac{15}{16}\)
Open
Question. 41
A bag has 24 balls: \(x\) red, \(2x\) white, \(3x\) blue. One ball drawn. Find probability it is (i) not red (ii) white.
Answer:
(i) \(\dfrac{5}{6}\), (ii) \(\dfrac{1}{3}\)
Open
Question. 42
Cards 1–1000 placed in a box. A player wins a prize if the card has a perfect square > 500. Players draw one card each without replacement. What is the probability that (i) the first player wins? (ii) the second player wins, if the first has already won?
Answer:
(i) \(\dfrac{9}{1000}\) ; (ii) \(\dfrac{8}{999}\)