Statistics & Probability Worked Examples for IGCSE Maths

Question

The table shows the heights of 50 plants. Height (cm): 0-10 (freq 5), 10-20 (freq 12), 20-30 (freq 18), 30-40 (freq 10), 40-50 (freq 5). Estimate the mean height.

Solution

1. 1

   Find the midpoint of each class

   Midpoints: 5, 15, 25, 35, 45

   Midpoint = (lower + upper) / 2 for each class.

2. 2

   Calculate midpoint times frequency for each class

   5x5=25, 15x12=180, 25x18=450, 35x10=350, 45x5=225

   Each midpoint represents all values in that class.

3. 3

   Sum the products and divide by total frequency

   Sum = 25+180+450+350+225 = 1230. Mean = 1230/50 = 24.6 cm

   Estimated mean = sum of (midpoint x frequency) / total frequency.

Question

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. Find the probability of getting one of each colour.

Solution

1. 1

   Draw the tree diagram branches

   First draw: P(R) = 4/10, P(B) = 6/10. Second draw after R: P(R) = 3/9, P(B) = 6/9. Second draw after B: P(R) = 4/9, P(B) = 5/9.

   Without replacement means the totals change for the second draw.

2. 2

   Find P(Red then Blue)

   P(RB) = 4/10 x 6/9 = 24/90

   Multiply along the RB branch.

3. 3

   Find P(Blue then Red)

   P(BR) = 6/10 x 4/9 = 24/90

   Multiply along the BR branch.

4. 4

   Add the two probabilities

   P(one of each) = 24/90 + 24/90 = 48/90 = 8/15

   OR means add: either RB or BR gives one of each colour.

Question

Draw a cumulative frequency curve and find the median and interquartile range. Masses (kg): 40-50 (freq 3), 50-60 (freq 7), 60-70 (freq 15), 70-80 (freq 12), 80-90 (freq 8), 90-100 (freq 5).

Solution

1. 1

   Calculate cumulative frequencies

   50: 3, 60: 10, 70: 25, 80: 37, 90: 45, 100: 50

   Add each frequency to the running total. Plot at upper class boundaries.

2. 2

   Find the median position

   Total = 50, so median is at 50/2 = 25th value

   The median is at the n/2 position on the cumulative frequency axis.

3. 3

   Read the median from the curve

   At cumulative frequency 25, read across to the curve and down to get approximately 70 kg

   Draw a horizontal line from 25 on the y-axis to the curve, then draw vertically down to read the x-value.

4. 4

   Find the quartiles and IQR

   LQ at 50/4 = 12.5th value, approximately 62 kg. UQ at 3(50)/4 = 37.5th value, approximately 80 kg. IQR = 80 - 62 = 18 kg.

   Lower quartile at n/4, upper quartile at 3n/4. IQR = UQ - LQ.

Question

In a class of 30 students, 18 play football (F), 15 play tennis (T), and 8 play both. Find P(a student chosen at random plays football or tennis).

Solution

1. 1

   Fill in the Venn diagram

   Both: 8. Football only: 18-8=10. Tennis only: 15-8=7. Neither: 30-10-8-7=5.

   Start from the intersection, then work outwards.

2. 2

   Find n(F union T)

   n(F union T) = 10 + 8 + 7 = 25

   Add all regions inside the circles.

3. 3

   Calculate the probability

   P(F union T) = 25/30 = 5/6

   Divide by the total number of students.