SPS SPS SM Statistics 2025 April — Question 4 8 marks

Exam BoardSPS
ModuleSPS SM Statistics (SPS SM Statistics)
Year2025
SessionApril
Marks8
TopicConditional Probability
TypeStandard Bayes with discrete events
DifficultyModerate -0.3 This is a straightforward conditional probability question using tree diagrams and basic probability rules. Part (a) requires simple multiplication along a branch, part (b) is standard Bayes' theorem application, and part (c) is a geometric distribution calculation. All techniques are routine A-level statistics content with clear problem structure and no novel insight required.
Spec2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles
A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that 15% of boards fail the mechanical inspection. Of those that pass the mechanical inspection, 8% fail the electrical test. Of those that fail the mechanical inspection, 60% fail the electrical test.
  1. If a board is randomly selected from production, what is the probability that it passes both inspections? [2]
  2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection? [3]
  3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections? [3]
A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that 15% of boards fail the mechanical inspection. Of those that pass the mechanical inspection, 8% fail the electrical test. Of those that fail the mechanical inspection, 60% fail the electrical test.

\begin{enumerate}[label=(\alph*)]
\item If a board is randomly selected from production, what is the probability that it passes both inspections? [2]

\item If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection? [3]

\item The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections? [3]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Statistics 2025 Q4 [8]}}
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