IGCSE Mathematics: Statistics and Probability (Extended) — Practice Questions (Cambridge 0580)

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — Cambridge International holds copyright in its own papers. Use these alongside the official past papers available from your board.

Each question practises a skill from the Extended (Supplement) content of the syllabus. After each answer there is a common mistake to watch out for.

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Questions

1. (Extended) A bag contains 5 red, 4 blue and 3 green counters. Two counters are taken at random without replacement. Given that the two counters are the same colour, find the probability that they are both red. [3]

2. (Extended) In a year group of 80 students, 45 study French, 30 study Spanish and 12 study both languages.

(a) A student who studies Spanish is chosen at random. Find the probability that this student also studies French.

(b) A student who studies at least one of the two languages is chosen at random. Find the probability that this student studies French only. [4]

3. (Extended) The probability that it rains on a given morning is 0.3. If it rains, the probability that Sami’s bus is late is 0.4. If it does not rain, the probability that his bus is late is 0.15.

(a) Find the probability that Sami’s bus is late on a given morning.

(b) Given that his bus is late, find the probability that it rained that morning. [4]

4. (Extended) The masses of some parcels are grouped into three classes: 0 < m ⩽ 20 kg has frequency 16, 20 < m ⩽ 30 kg has frequency 24 and 30 < m ⩽ 60 kg has frequency 27. Calculate the frequency density for each class, ready to draw a histogram. [3]

5. (Extended) In a histogram of waiting times t (minutes), the bar for 10 < t ⩽ 15 has a frequency density of 4.2 and the bar for 15 < t ⩽ 25 has a frequency density of 1.8. Assuming values are spread evenly within each class, estimate how many waiting times are in the range 12 < t ⩽ 20. [3]

6. (Extended) The times of 200 runners in a race are recorded. The cumulative frequencies are: 14 runners took 20 minutes or less, 52 took 25 minutes or less, 118 took 30 minutes or less, 170 took 35 minutes or less, 192 took 40 minutes or less and all 200 took 45 minutes or less.

(a) A cumulative frequency curve is drawn from these values. State the cumulative frequency at which you would read across to the curve to estimate the median, and the cumulative frequency you would use to estimate the upper quartile.

(b) How many runners took more than 35 minutes?

(c) Which class interval contains the median time? [4]

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Answers

1. (Extended) P(both red) = 5/12 × 4/11 = 20/132; P(both blue) = 4/12 × 3/11 = 12/132; P(both green) = 3/12 × 2/11 = 6/132 [1]. P(same colour) = 38/132 [1]. P(both red, given same colour) = 20/38 = 10/19 [1].

Common mistake: Giving P(both red) = 20/132 = 5/33 as the answer. “Given that” restricts you to the outcomes where both counters are the same colour, so divide by P(same colour).

2. (Extended) (a) 12 of the 30 Spanish students study French [1], so the probability is 12/30 = 2/5 [1].

(b) At least one language: 45 + 30 − 12 = 63 students [1]. French only: 45 − 12 = 33, so the probability is 33/63 = 11/21 [1].

Common mistake: Dividing by the whole year group of 80. When a condition is given, the denominator is the number of students who meet that condition.

3. (Extended) (a) P(late) = 0.3 × 0.4 + 0.7 × 0.15 [1] = 0.12 + 0.105 = 0.225 [1].

(b) P(rained, given late) = 0.12 ÷ 0.225 [1] = 8/15 (or 0.533, 3 s.f.) [1].

Common mistake: Giving 0.4 for part (b). That is the probability of being late given rain, which is the reverse of what is asked.

4. (Extended) Frequency density = frequency ÷ class width. 0 < m ⩽ 20: 16 ÷ 20 = 0.8 [1]. 20 < m ⩽ 30: 24 ÷ 10 = 2.4 [1]. 30 < m ⩽ 60: 27 ÷ 30 = 0.9 [1].

Common mistake: Dividing by the upper class boundary (for example 24 ÷ 30) instead of by the class width.

5. (Extended) From 12 to 15 minutes: 3 × 4.2 = 12.6 [1]. From 15 to 20 minutes: 5 × 1.8 = 9 [1]. Estimate = 12.6 + 9 = 21.6, so about 22 waiting times [1].

Common mistake: Reading the frequency density as if it were the frequency. Frequency = frequency density × width of the part of the class you need.

6. (Extended) (a) Median: read across from 100 (half of 200) [1]; upper quartile: read across from 150 (three-quarters of 200) [1].

(b) 200 − 170 = 30 runners [1].

(c) The 100th runner lies between the cumulative frequencies 52 and 118, so the median is in the class 25 < t ⩽ 30 minutes [1].

Common mistake: Reading off at the wrong height, for example using 100 for the upper quartile. For n values, use n/2 for the median, n/4 for the lower quartile and 3n/4 for the upper quartile.

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Where marks are usually lost

• Not reducing the sample space when a probability is “given that” something has happened.
• Dividing by the whole group instead of by the group that meets the condition.
• Confusing frequency with frequency density in histograms.
• Using class boundaries instead of class widths.
• Reading quartiles and the median at the wrong cumulative frequency.