In a hotel, the breakfast is a choice between yam (Y) or plantain (P) or both. The Venn diagram shows the choices made by 25 guests of the hotel.
(a) Find the value of x;
(b) What is the probability that a guest chosen at random chose only one of the two?
(a) 
\(U = 25\)
\(\therefore (2x + 1) + x + (x - 2)^{2} = 25\)
\(x^{2} - 4x + 4 + x + 2x + 1 = 25\)
\(x^{2} - x + 5 - 25 = 0\)
\(x^{2} - x - 20 = 0\)
\(\implies x^{2} - 5x + 4x - 20 = 0\)
\(x(x - 5) + 4(x - 5) = 0\)
\(\implies \text{x = -4 or 5}\)
But x cannot be negative in this case, hence, x = 5.
(b) Number of guests that selected just one meal = \(2x + 1 + x^{2} - 4x + 4\)
\(x^{2} - 2x + 5 = 5^{2} - 2(5) + 5 = 25 - 10 + 5 = 20\)
Probability of selecting a person who picked one meal = \(\frac{20}{25} = \frac{4}{5}\).
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