91Ó°ÊÓ

Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns in sets of elements. It is particularly useful when you want to figure out all possible combinations that can occur, like in Franz's vacation scenario. Here, we're using combinatorics to determine the different travel choices Franz's family has for their trip.

To apply combinatorics effectively:

• List out all the separate choices or steps involved. In this case, the choices are the destination country and the mode of travel.
• Combine each choice from one step with every choice from another step. This means combining each travel mode with each destination.

Therefore, when considering two destinations, Italy and Hungary, alongside their respective travel modes, you multiply the number of choices per step. For Italy: fly, drive, and train give three modes. For Hungary: drive and boat give two. Hence, the total number of combinations is 5, which forms the sample space: \[\{\text{(Italy, fly), (Italy, drive), (Italy, train), (Hungary, drive), (Hungary, boat)}\}\]

Event outcome analysis involves determining the specific outcomes or scenarios within the broader set of possibilities (the sample space) that fulfill a particular condition. It's about narrowing down the larger list to just those elements that meet the criteria of your specific event.

In Franz's situation, we are asked to analyze the event "fly to destination." The first step is to consider the sample space previously constructed. This sample space includes all possible travel mode and destination pairs.

• From the sample space \( \{ \text{(Italy, fly), (Italy, drive), (Italy, train), (Hungary, drive), (Hungary, boat)} \} \), look for outcomes where the travel mode is "fly."
• Quite straightforwardly, the only outcome satisfying this condition is traveling to Italy by flying, \((\text{Italy, fly})\).

This example demonstrates how event outcome analysis helps in isolating relevant outcomes that pertain to specific events of interest. It is a crucial skill when working with probability, as it simplifies complex problems into manageable parts.

Probability events refer to the occurrences or outcomes we're interested in when calculating the likelihood of something happening. These are subsets of the sample space that align with particular conditions — like Franz's family flying to their vacation destination.

Probability is calculated by dividing the number of favorable outcomes (those that match the event conditions) by the total number of possible outcomes, the entirety of the sample space.

• With the full sample space given as \( \{ \text{(Italy, fly), (Italy, drive), (Italy, train), (Hungary, drive), (Hungary, boat)} \} \), it contains 5 possible outcomes.
• The "fly to destination" event has only 1 favorable outcome: \((\text{Italy, fly})\).

Thus, the probability of this event occurring is \( \frac{1}{5} \), since there is only one event that involves flying to a destination out of the total five possible outcomes. Understanding these basic principles of probability allows us to make predictions and informed decisions based on likely outcomes.