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...
Question 1 Report
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?
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:
\[ P(\text{only one}) = \frac{20}{25} = \frac{4}{5} = 0.8 \]Download The App On Google Playstore
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