1. Find the mean of the distribution :
| Class | 1–3 | 3–5 | 5–7 | 7–10 |
|---|---|---|---|---|
| Frequency | 9 | 22 | 27 | 17 |
5.5
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 |
35
3. Calculate the mean of the following data :
| Class | 4–7 | 8–11 | 12–15 | 16–19 |
|---|---|---|---|---|
| Frequency | 5 | 4 | 9 | 10 |
\(\displaystyle 12.93\) (approx.)
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.
26 pages/day
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.
Rs 356.5
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.
109.92 seats
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.
123.4 kg
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?
14.48 km/l (Claim of 16 km/l is not supported.)
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.
| Less than | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 |
|---|---|---|---|---|---|---|---|---|
| Cumulative frequency | 4 | 8 | 21 | 26 | 32 | 37 | 39 | 40 |
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.
| 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 |
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 |
| 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 |
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 |
a=12, b=13, c=35, d=8, e=5, f=50
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.
(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 |
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.
| Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Frequency | 17 | 5 | 7 | 8 | 13 |
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.
≈ Rs 1263.16
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.
≈ 109.17 km/h
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.
≈ Rs 11,875
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.
≈ 201.7 g
Two dice are thrown. Find the probability of getting (i) the same number on both, (ii) different numbers.
(i) \(\dfrac{1}{6}\); (ii) \(\dfrac{5}{6}\)
Two dice are thrown. Probability that the sum is (i) 7 (ii) a prime number (iii) 1?
(i) \(\dfrac{1}{6}\), (ii) \(\dfrac{5}{12}\), (iii) \(0\)
Two dice are thrown. Probability that the product is (i) 6 (ii) 12 (iii) 7?
(i) \(\dfrac{1}{9}\), (ii) \(\dfrac{1}{9}\), (iii) \(0\)
Two dice are thrown and the product of the numbers is noted. Probability that the product is less than 9?
\(\dfrac{4}{9}\)
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).
| Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| Probability | \(\dfrac{1}{18}\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{18}\) |
A coin is tossed two times. Probability of getting at most one head?
\(\dfrac{3}{4}\)
A coin is tossed 3 times. List outcomes and find probability of (i) all heads (ii) at least two heads.
(i) \(\dfrac{1}{8}\), (ii) \(\dfrac{1}{2}\)
Two dice are thrown. Probability that the absolute difference of the numbers is 2?
\(\dfrac{2}{9}\)
A bag has 10 red, 5 blue, 7 green balls. Probability that a ball drawn is (i) red (ii) green (iii) not blue?
(i) \(\dfrac{5}{11}\), (ii) \(\dfrac{7}{22}\), (iii) \(\dfrac{17}{22}\)
From a deck, remove K, Q, J of clubs; draw one card from remaining. Probability that card is (i) a heart (ii) a king?
(i) \(\dfrac{13}{49}\), (ii) \(\dfrac{3}{49}\)
(Ref. Q28) Probability that the card is (i) a club (ii) 10 of hearts?
(i) \(\dfrac{10}{49}\), (ii) \(\dfrac{1}{49}\)
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.
(i) \(\dfrac{1}{10}\), (ii) \(\dfrac{3}{10}\), (iii) \(\dfrac{3}{5}\)
An integer is chosen between 0 and 100 (inclusive). Probability it is (i) divisible by 7 (ii) not divisible by 7?
(i) \(\dfrac{15}{101}\), (ii) \(\dfrac{86}{101}\)
Cards numbered 2 to 101 are in a box (100 cards). Probability that the card has (i) an even number (ii) a square number?
(i) \(\dfrac{1}{2}\), (ii) \(\dfrac{9}{100}\)
A letter of the English alphabet is chosen at random. Probability it is a consonant?
\(\dfrac{21}{26}\)
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?
\(\dfrac{69}{100}\)
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?
\(\dfrac{11}{75}\)
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.
(i) \(\dfrac{3}{4}\); (ii) \(\dfrac{5}{23}\)
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.
(i) \(\dfrac{4}{9}\), (ii) \(\dfrac{5}{9}\), (iii) \(\dfrac{1}{3}\), (iv) \(\dfrac{5}{18}\)
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.
(i) \(\dfrac{1}{8}\), (ii) \(\dfrac{1}{8}\), (iii) \(\dfrac{3}{4}\)
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?
(i) 6 totals (0,1,2,6,7,12); (ii) \(\dfrac{1}{3}\)
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?
(i) \(\dfrac{7}{8}\) ; (ii) \(\dfrac{15}{16}\)
A bag has 24 balls: \(x\) red, \(2x\) white, \(3x\) blue. One ball drawn. Find probability it is (i) not red (ii) white.
(i) \(\dfrac{5}{6}\), (ii) \(\dfrac{1}{3}\)
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?
(i) \(\dfrac{9}{1000}\) ; (ii) \(\dfrac{8}{999}\)