A corner shop's loyalty app has \(50\) registered customers. Set \(A\) is customers who buy coffee, \(n(A)=28\); set \(B\) is customers who buy pastries, \(...
Question 1 Report
A corner shop's loyalty app has \(50\) registered customers. Set \(A\) is customers who buy coffee, \(n(A)=28\); set \(B\) is customers who buy pastries, \(n(B)=24\); \(6\) customers buy neither, shown on the Venn diagram.
- Form an equation in \(x\), using the total of \(50\) customers, and solve it. (2)
- Find the probability that a randomly selected customer buys pastries only. (1)
- Write down \(n(A \cup B)\). (1)
Answer Details
Every one of the 50 customers falls into exactly one region of the Venn diagram, so adding all four regions and setting the total to 50 gives an equation for \(x\); the completed diagram then answers the remaining parts directly.
(a) Adding the four regions: \((28-x)+x+(24-x)+6=50\), which simplifies to \(58-x=50\), so \(x=8\), giving the completed regions 20, 8, 16 and 6 shown above. [2 marks]
(b) "Pastries only" is the region inside \(B\) but outside \(A\): \(24-x=24-8=16\) customers, so \(P(\text{pastries only}) = \dfrac{16}{50} = \dfrac{8}{25}\). [1 mark]
(c) \(n(A\cup B)\) is every region inside at least one circle: \(20+8+16=44\). [1 mark]
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