91Ó°ÊÓ

Understanding probability helps us determine how likely an event is to occur. In this exercise, the event in question is whether Yolanda will host the first party and Lorrie the second.

To calculate the probability of an event, we use the formula: \( P(event) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ possible\ outcomes} \)

Here, the 'favorable outcome' is Yolanda hosting the first and Lorrie hosting the second. The 'total possible outcomes' include all combinations of who can host the first and second parties.

Using this approach, we determine that the probability of Yolanda and Lorrie hosting sequentially is \( \frac{1}{20} \). This formula is essential for understanding various scenarios where certain outcomes are preferred over others.

Combinatorics is the field of mathematics that deals with counting combinations and arrangements of objects. In this problem, it's used to figure out how many different ways two homes can be selected from five for hosting the party.

Consider that when choosing the first home, there are 5 different choices. After one home is chosen, only 4 homes remain available for the second choice. Using this logic, the total number of combinations is calculated as: \[ 5 \times 4 = 20 \]

This means there are 20 different possible hosting sequences. Combinatorics often involves factorial calculations and other techniques to determine possible arrangements, but here a simple multiplication suffices. This field is crucial in many aspects of probability and statistics.

Random selection ensures that each subject or item has an equal chance of being chosen. In our problem, selecting the hosts randomly means there is no favoritism or predetermined sequence.

When Yolanda's home is randomly chosen among the five, her chance of being selected first is \( \frac{1}{5} \). After she's chosen, Lorrie's chance of being selected from the remaining four homes is \( \frac{1}{4} \).

These probabilities combine to reflect the event's likelihood: \[ P(Yolanda \ first \ and \ Lorrie \ second) = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20} \]

Random selection is a fundamental concept in probability theory, ensuring fairness and unbiased outcomes in various processes.