A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour:
10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets
- (i) Calculate the probability that a randomly picked sweet from the sample is green.
- (ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
Step-by-step solution
Idea: The sample gives a relative frequency for each colour. Use it as the probability for the sample, and as the estimated share of that colour in the whole bag.
(i) Calculate the probability that a randomly picked sweet from the sample is green.
- Total in the sample: 10 + 8 + 7 + 5 = 30. Green sweets: 8.½ mark
- P(green) = Number of green sweetsTotal sweets in the sample = 830 = 415 ≈ 0.267 (about 27%).½ mark
(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
- In the sample, P(yellow) = 730.½ mark
- The sample was picked at random, so the whole bag should have about the same share of yellow sweets.½ mark
- Estimated yellow sweets = 730 × 600 = 7 × 20 = 140.1 mark
Check: Scale the whole sample up by 600 ÷ 30 = 20: red 200, green 160, yellow 140, blue 100. Total 600 ✓.
Answer to write in the exam
(i)
Total sweets in sample = 10 + 8 + 7 + 5 = 30; green = 8
∴ P(green) = 830 = 415 ≈ 0.27
(ii)
P(yellow) in sample = 730
Estimated yellow sweets = 730 × 600 = 140
∴ About 140 sweets are likely to be yellow
Common mistakes that cost marks
- Writing 822 for P(green), comparing green with the other colours instead of with the whole sample of 30.
- In (ii), dividing 600 by 7, or using 7600. The share of yellow (730) must be multiplied by the total, 600.
- Saying there are exactly 140 yellow sweets. It is an estimate from a sample, so write “about 140”.
How this can come in the exam
A fruit seller picks 50 mangoes at random from a crate of 1000 and checks them: 36 are ripe, 10 are unripe and 4 are spoilt.
(a) Find the probability that a mango from the sample is ripe. (1)
(b) Find the probability that a mango from the sample is not spoilt. (1)
(c) Estimate the number of spoilt mangoes in the crate. (2)
Show answer
(a) 3650 = 0.72.(b) (50 − 4)/50 = 4650 = 0.92.
(c) P(spoilt) = 450 = 0.08; estimate = 0.08 × 1000 = 80 mangoes.
In a random sample of 20 beads from a jar, 6 are white. The jar has 500 beads. The best estimate of the number of white beads in the jar is
- 6
- 100
- 150
- 300
Show answer
(C) 150
620 × 500 = 0.3 × 500 = 150.
Try one yourself
In a random sample of 40 pencils from a box of 400, 12 are blunt. (a) Find the probability that a pencil from the sample is blunt. (b) Estimate the number of blunt pencils in the box.
Show answer
(a) 1240 = 0.3. (b) 0.3 × 400 = 120 pencils.
More questions like this
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14 students: Science Club | 11 students: Arts Club |
9 students: Sports Club | 6 students: Debate Club
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