People's Education Press: High School Mathematics, Compulsory Course II (Version A): Set-Theoretic Expression of Random Experiments and Sample Spaces

Random Experiment (Random Trial): We refer to the realization and observation of a random phenomenon as a random experiment, commonly abbreviated as 'experiment', denoted by the letter $E$. Each possible outcome in the experiment is calledsample point (Sample Point), and the set of all sample points is known assample space (Sample Space), typically represented by $\Omega$.

Core Concept Analysis

In probability theory, we use set language to describe random phenomena. If an experiment has only a finite number of possible outcomes, it is referred to asfinite sample space. For example:

Tossing a coin: $\Omega = \{h, t\}$
Tossing two coins: $\Omega = \{(\text{Head, Head}), (\text{Head, Tail}), (\text{Tail, Head}), (\text{Tail, Tail})\}$

Moreover, statistical inference is highly significant in real-world applications, such asBody Mass Index (BMI) research. The Chinese adult standards are: $BMI < 18.5$ indicates underweight; $18.5 \le BMI < 24$ is normal; $24 \le BMI < 28$ is overweight; $BMI \ge 28$ is obese.

Samples are inherently random; thus, statistical inferences drawn from samples to estimate populations carry uncertainty. This is a crucial consideration when interpreting statistical results in real-world contexts.

$$BMI=\frac{\text{Weight (kg)}}{\text{Height}^2 (\text{m}^2)}$$

QUESTION 1

How many sample points are there in the sample space $\Omega$ of tossing a fair six-sided die?

QUESTION 2

Which of the following statements about random experiments is correct?

The outcome of the experiment is predetermined before it occurs

The set of all sample points is called the sample space

A sample point is a random event

There are only 2 sample points when tossing a coin twice

QUESTION 3

Write the sample space for the random experiment of recording a student’s gender.

$\Omega = \{\text{Male}, \text{Female}\}$

$\Omega = \{\text{Student}\}$

$\Omega = \{\text{Male}\}$

$\Omega = \{1, 0\}$

QUESTION 4

In a family with two children, what is the sample space $\Omega$ when observing the genders of both children?

{(Male, Male), (Female, Female)}

{(Male, Male), (Male, Female), (Female, Male), (Female, Female)}

{2 males, 1 male & 1 female, 2 females}

{(Male, Female)}

QUESTION 5

A person shoots at a target 3 times; observe the number of hits. What is the sample space for this experiment?

{Hit, Miss}

{0, 1, 2, 3}

{1, 2, 3}

{0, 3}

QUESTION 6

During a lottery draw, one ball is drawn from 10 balls labeled 0–9. How many possible outcomes are there for this experiment?

Infinitely many

QUESTION 7

For a parallel circuit (components A and B in parallel), which sample points belong to event N = 'circuit is open'? (1 represents normal, 0 represents failure)

{(1, 1)}

{(1, 0), (0, 1)}

{(0, 0)}

{(1, 1), (1, 0), (0, 1)}

QUESTION 8

Which of the following statements about BMI is incorrect?

BMI stands for Body Mass Index

BMI = Weight / Height

BMI ≥ 28 indicates obesity

18.5 ≤ BMI < 24 is normal

QUESTION 9

What is the probability of getting 'one head and one tail' when tossing two coins?

0.25

0.5

0.75

QUESTION 10

If the median of a dataset is much smaller than the mean, what might exist in the data?

Outlier small values

Outlier large values

Mode

Uniform frequency distribution

Comprehensive Case Study and Logical Challenge

Task 1

[Writing Task] Employee BMI Statistical Analysis Report

Based on BMI data from 90 male and 50 female employees (men: 23.5, 21.6, 30.6...; women: 21.8, 18.2, 25.2...), write a statistical report. Word count requirement: at least 200 words.

Reference Model:
1. Data Presentation: Suggest using frequency distribution histograms to show male and female employee BMI distributions separately, or box plots for comparison. Based on the data, male employees’ average BMI is approximately 24.2, and female employees’ is around 22.5.
2. Comparison of Differences: Male employees have a significantly higher proportion of overweight individuals (BMI ≥ 24), and obesity (BMI ≥ 28) is predominantly observed among males. Female employees mostly fall within the normal range, with some showing underweight conditions.
3. Overall Analysis: The overall health status of employees is acceptable, but the male group faces a higher risk of overweight, possibly due to prolonged sitting at work or lack of physical activity.
4. Recommendations: The company could introduce stretching exercises during tea breaks, label dish calorie counts in the cafeteria, and regularly organize badminton or running events to encourage male employees to manage their weight.

Task 2

Review of Statistical Fundamentals

Briefly explain: (1) What information does a frequency distribution histogram provide? (2) What are the characteristics of mean, median, and mode? (3) What do variance and standard deviation measure?

Reference Answers:
(1) Histogram: It allows for intuitive observation of data central tendency, spread, and distribution shape (e.g., symmetry).
(2) Central Tendency: The mean reflects the average level and is sensitive to outliers; the median is the middle value and is robust against anomalies; the mode reflects the most frequently occurring data point.
(3) Dispersion: Variance and standard deviation reflect the magnitude of data variation. Larger values indicate greater deviation from the center and higher instability.

Task 3

Is the two-coin game fair?

Game rules: Both coins show heads or both show tails → Player A wins; one head and one tail → Player B wins. Judge and explain your reasoning.

Solution:
The game is fair.
The sample space $\Omega = \{(h, h), (h, t), (t, h), (t, t)\}$ contains 4 sample points.
Player A’s winning event $A = \{(h, h), (t, t)\}$ includes 2 sample points, so $P(A) = 2/4 = 0.5$.
Player B’s winning event $B = \{(h, t), (t, h)\}$ includes 2 sample points, so $P(B) = 2/4 = 0.5$.
Since $P(A) = P(B)$, the game is fair.

Task 4

Discussion on Frequency vs. Probability

‘Using the frequency $f_n(A)$ of event A’s occurrence to estimate its probability $P(A)$, the larger the number of repeated trials $n$, the more accurate the estimate.’ Is this statement correct? Provide an example.

Solution:
This statement is correct. As the number of trials $n$ increases, the frequency $f_n(A)$ of a random event’s occurrence becomes stable and gradually approaches its probability $P(A)$.
Example: Tossing a fair coin. After 10 tosses, you might get 7 heads (frequency 0.7); after 1,000 tosses, the number of heads usually fluctuates around 500 (frequency close to 0.5); after 100,000 tosses, the frequency stabilizes very closely around 0.5. This illustrates the law of large numbers intuitively.