Year 8 Probability and statistics: free practice test with answers

Sample probability and statistics questions with answers

Try each one before you open the answer.

1. Interpreting graphs and tables

   The table shows the rainfall in a town for the first six months of a year.

   Month	Jan	Feb	Mar	Apr	May	Jun
   Rainfall (mm)	42	38	55	61	74	69

   (a) Which month had the most rain?

   (b) What was the total rainfall for the six months?

   Show the answer

   Key idea: Read values straight from the table, then add them for the total.

   1. (a) The largest value is $74$ mm, in May
   2. (b) $42 + 38 + 55 + 61 + 74 + 69 = 339$ mm

   Answer: (a) May (b) $339$ mm

2. Frequency tables and tallies

   Twenty students were asked how many pets they have. Their answers were:

   $$0, 1, 2, 1, 0, 3, 1, 2, 2, 1, 0, 1, 4, 1, 2, 0, 1, 3, 2, 1$$

   (a) Make a frequency table for the data.

   (b) What is the relative frequency of students with exactly $1$ pet?

   Show the answer

   Key idea: Count how many times each value occurs (a tally helps). Relative frequency $= \dfrac{\text{frequency}}{\text{total}}$.

   1. (a) Count each value: $0$ appears $4$ times, $1$ appears $8$ times, $2$ appears $5$ times, $3$ appears $2$ times and $4$ appears once
   2. (a) Check the total: $4 + 8 + 5 + 2 + 1 = 20$ ✓
   3. (b) $\dfrac{8}{20} = 0.4$

   Answer: (a) $0$: $4$, $1$: $8$, $2$: $5$, $3$: $2$, $4$: $1$ (b) $0.4$

3. Graphs of frequency tables

   The ages of $10$ people at a party are:

   $$23, 35, 31, 27, 42, 38, 29, 35, 46, 31$$

   (a) Draw an ordered stem-and-leaf plot of the ages.

   (b) Use your plot to find the median age.

   Show the answer

   Key idea: In a stem-and-leaf plot the tens digit is the stem and the ones digit is the leaf. An ordered plot lists the data in order, which makes the median easy to find.

   1. (a) Stem $2$: $23, 27, 29$; stem $3$: 31, 31, 35, 35, 38; stem $4$: $42, 46$ (key: $2 \mid 3$ means $23$)
   2. (b) There are $10$ values, so the median is halfway between the 5th and 6th values
   3. (b) The 5th value is $31$ and the 6th is $35$, so the median is $\dfrac{31 + 35}{2} = 33$

   Answer: (a) $2 \mid 3\ 7\ 9$; $3 \mid 1\ 1\ 5\ 5\ 8$; $4 \mid 2\ 6$ (b) $33$

4. Measures of spread

   The daily maximum temperatures, in degrees Celsius, in Mildura for one week were:

   $$34, 41, 29, 38, 45, 31, 40$$

   Find the range.

   Show the answer

   Key idea: Range $=$ highest value $-$ lowest value.

   1. Highest $= 45$, lowest $= 29$.
   2. Range $= 45 - 29 = 16$.

   Answer: $16^\circ\text{C}$

5. Surveying and sampling

   A school of 800 students wants to find out the most popular sport among its students. The school captain surveys every 10th student on the school roll. State (a) the population, (b) the sample, and (c) the sampling method used.

   Show the answer

   Key idea: The population is the entire group being studied. The sample is the smaller group actually surveyed. The sampling method describes how the sample was chosen.

   1. The study is about students at this school, so the population is all 800 students at the school.
   2. Only the students selected from the roll are surveyed, so the sample is the students chosen (every 10th student on the roll).
   3. Students are selected at a fixed interval (every 10th), so the sampling method is systematic sampling.

   Answer: (a) All 800 students at the school. (b) Every 10th student on the school roll. (c) Systematic sampling.

6. Probability

   A bag contains 20 marbles. The probability of picking a red marble at random is $\dfrac{3}{20}$. Find the probability of not picking a red marble.

   Show the answer

   Key idea: Complementary events cover all possible outcomes, so $P(\text{not A}) = 1 - P(A)$.

   1. Identify the given probability: $P(\text{red}) = \dfrac{3}{20}$
   2. Apply the complement rule:\

   $$P(\text{not red}) = 1 - \dfrac{3}{20}$$

   3. Subtract:\

   $$P(\text{not red}) = \dfrac{20}{20} - \dfrac{3}{20} = \dfrac{17}{20}$$

   Answer: $\dfrac{17}{20}$

7. Tree diagrams

   A bag contains one red marble (R) and one blue marble (B). A marble is drawn, its colour is recorded, and it is put back. A second marble is then drawn. List all possible outcomes using a tree diagram and find the probability of getting two red marbles.

   Show the answer

   Key idea: Draw a tree diagram by listing every branch for the first draw, then every branch for the second draw from each first-draw branch. Because the marble is replaced, the bag is the same for both draws.

   1. First draw has two branches: R and B.
   2. From each branch, the second draw also has two branches: R and B. The complete sample space is: RR, RB, BR, BB, 4 equally likely outcomes.
   3. Only one outcome is two red marbles: RR.
   4. $P(\text{RR}) = \frac{1}{4}$

   Answer: $P(\text{two red marbles}) = \dfrac{1}{4}$

8. Venn diagrams and two-way tables

   In a class of 30 students, 18 students play football, 14 students play basketball, and 6 students play both sports. A student is chosen at random. Find the probability that the student plays neither football nor basketball.

   Show the answer

   Key idea: Draw a Venn diagram with two overlapping circles, one for football (F) and one for basketball (B), inside a rectangle representing all 30 students. Place the overlap count first, then fill in the rest of each circle.

   1. The overlap (both sports) contains 6 students.
   2. Football only: $18 - 6 = 12$ students. Basketball only: $14 - 6 = 8$ students.
   3. Total accounted for inside the circles: $12 + 6 + 8 = 26$ students.
   4. Students outside both circles (neither sport): $30 - 26 = 4$ students.
   5. $P(\text{neither}) = \frac{4}{30} = \frac{2}{15}$

   Answer: $P(\text{neither football nor basketball}) = \dfrac{2}{15}$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 8 integer operations worksheet
• Year 8 statistics worksheet
• Year 8 complementary events and probability worksheet
• Year 8 probability worksheet