A car workshop keeps records of warranty repairs. Of the 60 cars repaired last month, 24 had an engine fault, 18 had an electrical fault, and 8 had both. Th...
Question 1 Report
A car workshop keeps records of warranty repairs. Of the 60 cars repaired last month, 24 had an engine fault, 18 had an electrical fault, and 8 had both. The Venn diagram shows the number with both faults; the other regions are blank.
- Work out the number with neither fault, and complete the Venn diagram. (2)
- A car is chosen at random. Work out the probability it had exactly one fault. (2)
- Given that a car had an electrical fault, work out the probability it also had an engine fault. (2)
Answer Details
This question tests completing a two-circle Venn diagram from overlap information, then using the completed regions for simple and conditional probability.
(a) The 24 cars with an engine fault include the 8 with both faults, so \( 24 - 8 = 16 \) cars have an engine fault only. Similarly, \( 18 - 8 = 10 \) cars have an electrical fault only. The three regions inside the circles (16, 8 and 10) account for \( 16 + 8 + 10 = 34 \) cars, so the number with neither fault is:
\[ 60 - 34 = 26 \] [2 marks]
(b) "Exactly one fault" means only engine or only electrical, not both, so add the two "only" regions and divide by the total:
\[ P(\text{exactly one fault}) = \frac{16 + 10}{60} = \frac{26}{60} = \frac{13}{30} \] [2 marks]
(c) "Given an electrical fault" restricts attention to the electrical circle only (18 cars), so divide the overlap (8, which have both faults) by the electrical total:
\[ P(\text{engine} \mid \text{electrical}) = \frac{8}{18} = \frac{4}{9} \] [2 marks]
The key Venn-diagram skill in part (a) is subtracting the overlap from each individual total before adding, so that the 8 cars with both faults are not counted twice.
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