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 t...

Answer Details

From the Venn diagram the three disjoint regions are:

• Plantain only (\(P\) region outside the overlap): \(2x+1\)
• Both yam and plantain (the overlap): \(x\)
• Yam only (\(Y\) region outside the overlap): \((x-2)^2\)

Every one of the 25 guests chose yam, plantain, or both, so the three regions add up to the total number of guests.

(a) Finding the value of \(x\)

\[ (2x+1) + x + (x-2)^2 = 25 \]

Expand \((x-2)^2 = x^2 - 4x + 4\):

\[ 2x + 1 + x + x^2 - 4x + 4 = 25 \]\[ x^2 - x + 5 = 25 \]\[ x^2 - x - 20 = 0 \]

Factorising:

\[ (x-5)(x+4) = 0 \]\[ x = 5 \quad \text{or} \quad x = -4 \]

A number of guests cannot be negative, so we reject \(x = -4\).

\[ \boxed{x = 5} \]

(b) Probability that a guest chose only one of the two

Substitute \(x = 5\) into each region:

\[ \text{Plantain only} = 2(5)+1 = 11 \]\[ \text{Both} = 5 \]\[ \text{Yam only} = (5-2)^2 = 9 \]

Check: \(11 + 5 + 9 = 25\), which matches the total number of guests.

Guests who chose only one of the two are those in "plantain only" or "yam only":

\[ 11 + 9 = 20 \]

Therefore the required probability is:

\[ P(\text{only one}) = \frac{20}{25} = \frac{4}{5} = 0.8 \]