Year 9 Venn Diagrams and Two-Way Tables Worksheets & Practice

Worked examples

Year 9 venn diagrams and two-way tables: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Reading probabilities from a two-way table

   A group of 40 students were surveyed about whether they play sport and whether they play a musical instrument. The results are shown in the two-way table below.

   	Plays sport	Does not play sport	Total
   Plays instrument	9	6	15
   Does not play instrument	17	8	25
   Total	26	14	40

   Use the table to estimate: (a) $P(\text{plays sport and plays instrument})$ (b) $P(\text{plays sport or plays instrument})$, using inclusive or (c) $P(\text{does not play sport})$

   Show worked solution

   Key idea: Relative frequency $= \dfrac{\text{frequency of event}}{\text{total number of observations}}$ estimates probability. For inclusive or, use $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

   1. There are 40 students in total.
   2. (a) 9 students play both sport and an instrument.
      $P(\text{sport and instrument}) = \frac{9}{40}$
   3. (b) 26 play sport, 15 play instrument, 9 play both.
      $P(\text{sport or instrument}) = \frac{26}{40} + \frac{15}{40} - \frac{9}{40} = \frac{32}{40} = \frac{4}{5}$
   4. (c) 14 students do not play sport.
      $P(\text{not sport}) = \frac{14}{40} = \frac{7}{20}$

   Answer: (a) $\dfrac{9}{40}$, (b) $\dfrac{4}{5}$, (c) $\dfrac{7}{20}$

2. Example 2 · Building and using a Venn diagram

   In a class of 30 students, 18 have a library card, 14 have a bus pass, and 7 have both. Use this information to construct a Venn diagram and then find: (a) $P(\text{library card only, and not bus pass})$ (b) $P(\text{library card or bus pass, but not both})$, exclusive or (c) $P(\text{neither library card nor bus pass})$

   Show worked solution

   Key idea: Fill in a Venn diagram from the inside out. Exclusive or means one event or the other, but not both.

   1. Both: 7. Library card only: $18 - 7 = 11$. Bus pass only: $14 - 7 = 7$. Neither: $30 - 11 - 7 - 7 = 5$.
   2. (a) Library card only (not bus pass): 11 students.
      $P(\text{library only}) = \frac{11}{30}$
   3. (b) Exclusive or means library only or bus pass only.
      $P(\text{library or bus pass, not both}) = \frac{11 + 7}{30} = \frac{18}{30} = \frac{3}{5}$
   4. (c) Neither: 5 students.
      $P(\text{neither}) = \frac{5}{30} = \frac{1}{6}$

   Answer: (a) $\dfrac{11}{30}$, (b) $\dfrac{3}{5}$, (c) $\dfrac{1}{6}$

3. Example 3 · Real-life problem: estimating probabilities from survey data

   A café owner surveys 60 customers on a Saturday about their orders. 35 customers order coffee, 28 order food, and 15 order both coffee and food. One of the 60 customers is chosen at random.

   (a) Estimate the probability that the chosen customer ordered coffee or food (inclusive or). (b) Estimate the probability that the chosen customer ordered coffee or food, but not both (exclusive or). (c) The café owner says: "More than half of my customers order neither coffee nor food on Saturdays." Is this claim correct? Justify your answer using probability.

   Show worked solution

   Key idea: Use relative frequencies from the survey as probability estimates. Organise the data with a Venn diagram, then apply the or rules.

   1. Coffee only: $35 - 15 = 20$. Food only: $28 - 15 = 13$. Both: 15. Neither: $60 - 20 - 13 - 15 = 12$.
   2. (a) Inclusive or (coffee or food or both):
      $P(\text{coffee or food}) = \frac{20 + 13 + 15}{60} = \frac{48}{60} = \frac{4}{5}$
   3. (b) Exclusive or (coffee only or food only):
      $P(\text{coffee or food, not both}) = \frac{20 + 13}{60} = \frac{33}{60} = \frac{11}{20}$
   4. (c) Customers who ordered neither: 12.
      $P(\text{neither}) = \frac{12}{60} = \frac{1}{5} = 0.2$ Since $0.2 < 0.5$, the claim is incorrect. Only $\frac{1}{5}$ of customers ordered neither, which is well below half.

   Answer: (a) $\dfrac{4}{5}$, (b) $\dfrac{11}{20}$, (c) The claim is incorrect; only $\dfrac{1}{5}$ of customers ordered neither.

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