91Ó°ÊÓ

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It's about predicting future events based on known possibilities. For example, in our junior high school dental survey, we want to know the chance that the first student with a cavity is the third one examined.
The probability of an event happening is given by a fraction where:

• The numerator represents the number of successful outcomes
• The denominator represents the total number of possible outcomes

In our exercise, the probability that a student has a cavity is \(\frac{2}{3}\). The probability of a student not having a cavity is \(\frac{1}{3}\).
When dealing with multiple events, we need to multiply the probabilities of each individual event occurring.

The binomial probability deals with scenarios where there are exactly two possible outcomes, like success or failure. In our dental survey exercise, each student either has a cavity or doesn't. We can use the binomial probability formula here.
The general formula for binomial probability is given by: \[ P(X = k) = {n \choose k} \cdot p^k \cdot (1-p)^{n-k} \] where:

• \(n\) is the total number of trials (or students)
• \(k\) is the number of successful trials (students with cavities)
• \(p\) is the probability of a success (cavity)

In our case, we're interested in the specific order: the first student having a cavity on the third try. This isn't a standard binomial problem since it requires a specific sequence of events. Instead, we calculate it by sequentially multiplying individual probabilities: \[ P(C^{'} ) \times P(C^{'} ) \times P(C) = \frac{1}{3} \times \frac{1}{3} \times \frac{2}{3} = \frac{1}{27} \]

Basic statistics helps us understand and interpret data. In the context of our problem, we use several fundamental concepts:

• Mean: The average of a set of numbers, not directly relevant to our probability question but important in general data analysis.
• Median: The middle value in a dataset when arranged in ascending order.
• Mode: The most frequently occurring value in a dataset.

For our junior high school dental survey, we're particularly interested in probability, a core part of statistics. Understanding the likelihood of an event — like finding the first cavity on the third student examined — helps us make informed predictions. By breaking the problem down into steps and applying basic statistical and probability concepts, we determine that the probability is \( \frac{1}{27} \).
Statistics is not only about complex calculations but also about logical and critical thinking to solve real-life problems.