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Probability, STD2 2026 HSC 32

At a particular school, the probability that a student walks to school is \(\dfrac{3}{10}\) and the probability that a student wears the correct school uniform is \(\dfrac{9}{10}\).

The probability tree shows this information.
 

  1. One student at the school is selected at random.
  2. What is the probability that this student walks to school and is not wearing the correct school uniform?   (1 mark)
  3. Two students at the school are selected at random.
  4. What is the probability that at least one of these students walks to school and is not wearing the correct school uniform?   (2 marks)
Show Answers Only

a.    \(\dfrac{3}{100}\)

b.    \(\dfrac{591}{10\ 000}\)

Show Worked Solution

a.    \(P(\text{W and NC})=\dfrac{3}{10}\times\dfrac{1}{10}=\dfrac{3}{100}\)
 

b.    \(\text{Let E = student walks and is not in correct uniform}\)

\(P(\text{E})=\dfrac{3}{100}\ \ \Rightarrow\ \ P(\text{not E})=1-\dfrac{3}{100}=\dfrac{97}{100}\)

\(\text{“At least one E” is the complement of “neither student is E”.}\)

\(P(\text{at least one E})\) \(=1-P(\text{not E})\times P(\text{not E})\)
  \(=1-\dfrac{97}{100}\times\dfrac{97}{100}\)
  \(=1-\dfrac{9409}{10\ 000}\)
  \(=\dfrac{591}{10\ 000}\)