- The combined area of the food-truck and games zones is \(8,000 + 10,000 = 18,000\) sq ft.
- The probability is the ratio of this combined area to the total area: \(P = \frac{18,000}{54,000}\).
- Simplifying the fraction gives the final result: \(\boxed{\frac{1}{3}}\).
Explanation
- Analyze the problem and given data
We are given the total area of the fairground, which is \(A_{total} = 54,000\) square feet. Within this fairground, there are two distinct zones: the food-truck zone with an area of \(A_{food} = 8,000\) square feet, and the games zone with an area of \(A_{games} = 10,000\) square feet. We are told these two zones do not overlap. We need to find the probability that a balloon landing at a random point in the fairground lands in either the food-truck zone or the games zone.
- Calculate the combined area of the zones of interest
Since the zones do not overlap, the total area covered by both zones is simply the sum of their individual areas. We calculate this combined area as:
\[A_{combined} = A_{food} + A_{games} = 8,000 + 10,000 = 18,000 \text{ square feet}\]
- Calculate the probability
The probability \(P\) of an event occurring in a specific region, assuming a uniform distribution of landing points, is the ratio of the area of that region to the total area of the fairground. Thus, the probability is:
\[P = \frac{A_{combined}}{A_{total}} = \frac{18,000}{54,000}\]
To simplify this fraction, we can divide both the numerator and the denominator by \(18,000\):
\[P = \frac{18,000 \div 18,000}{54,000 \div 18,000} = \frac{1}{3}\]
Thus, the probability is \(\frac{1}{3}\).
Examples
This type of calculation is very useful for spatial planning and risk assessment. For instance, if you are designing a city park and want to know the probability of a bird landing in a specific area (like a pond or a playground), you can compare the area of those features to the total area of the park. If a park has a total area of \(100,000\) square meters and a pond that is \(20,000\) square meters, the probability of a bird landing in the pond is \(\frac{20,000}{100,000} = \frac{1}{5}\).