The essential differences between random phenomena and deterministic phenomena
To understand random phenomena, it is necessary to first clarify the core boundary between them and deterministic phenomena:
A random phenomenon is a "contingency phenomenon whose result cannot be predicted under the same conditions" - even if all known variables are strictly controlled, multiple different results will still occur in repeated observations. For example, when making two cups of coffee with the same coffee machine, the taste may vary slightly; when sowing the same batch of seeds in the same field, the final yields will also vary.
A deterministic phenomenon is a necessity phenomenon where the result is uniquely determined once the conditions are given. As long as the conditions are satisfied, the result can be accurately predicted with 100% certainty. For example, water boils at 100°C under standard atmospheric pressure and the sum of the interior angles of a triangle is 180°. These results will not change due to any random factors.
Judgment of random phenomena: Dissect the logic with example questions
We verified the differences between the two types of phenomena through four specific cases. The key lies in "whether the results can be predicted with 100% certainty":
A. The sun rises in the east: The Earth's rotation from west to east is a fixed law. As long as the conditions of the Earth's revolution and rotation remain unchanged, the sun will definitely rise in the east — there is no uncertainty in the result, and it belongs to a deterministic phenomenon.
Objects thrown upward will definitely fall: Due to the inevitable effect of the Earth's gravity, no matter what objects are thrown (as long as they are on the Earth's surface), they will eventually fall - the result is bound to occur, which belongs to a deterministic phenomenon.
C Tomorrow's maximum temperature: Weather forecasts are probability predictions based on models. The actual temperature will still be affected by random factors such as real-time atmospheric movements and cloud cover changes (for example, a sudden gust of wind may lower the temperature by 2°C). It cannot be predicted with complete accuracy and is a random phenomenon.
D. Weight of newborn babies: There are differences in each baby's genes, maternal nutrition, and the delivery process. Some babies weigh 3.5 kg, while others weigh 2.8 kg. The results are accidental and belong to random phenomena.
Random phenomena in quality management: Ubiquitous uncertainties
1. Future market share of new products: It is affected by various variables such as market feedback (whether users like it), competitors' actions (whether opponents reduce prices), and promotional effects (whether advertisements reach the target audience). Even if a perfect market research is conducted, the final market share may still fluctuate from 10% to 50% — there is no necessarily successful outcome.
2. Errors in machining machine shafts: The slight vibration of the machine tool, the wear of the cutting tool, and the hardness difference of the material will all cause the actual size of the shaft to deviate from the design value. For example, if the designed diameter is 10 mm, the actual diameter may be 9.98 mm or 10.02 mm, and the error is different for each machining.
3. Time of the first failure of a TV set: The wear - and - tear rate of parts, the user's frequency of use (for example, some people watch TV for 10 hours a day while others only watch for 1 hour), and the ambient temperature (such as placing the TV in the sun on the balcony vs. placing it in a shady area in the living room) will all affect the failure time. For TV sets of the same model, some may break down after 5 years while others may break down after 1 year, and the failure time is unpredictable.
4. Weight of luncheon meat: Even if the filling equipment on the production line has been calibrated, slight variations in the weight of each can may still occur due to fluctuations in the raw material density (such as changes in the humidity of the minced pork) and changes in the filling flow rate (such as minor fluctuations in the pipeline pressure). For products labeled "340g", the actual weight may randomly fluctuate between 338g and 342g.
Sample points and sample spaces: The "language" for describing random phenomena
To systematically analyze random phenomena, two basic concepts need to be clarified first:
Sample point: The smallest indivisible result of a random phenomenon (such as "heads" and "tails" when tossing a coin, and "1 dot", "2 dots" when rolling a die) — these results cannot be split into smaller units.
Sample space: The set of all sample points (denoted by Ω), representing "all possible outcomes" of a random phenomenon.
We use four examples to illustrate the specific forms of sample points and sample spaces:
Coin toss: The sample points are "heads" and "tails", and the sample space Ω = {heads, tails} (two discrete points).
Rolling a die: The sample points are "1 dot", "2 dots", ..., "6 dots". The sample space Ω = {1, 2, 3, 4, 5, 6} (six discrete points).
The time of the first failure of a TV set: The sample point is t hours (t ≥ 0, such as 100 hours, 200.5 hours), and the sample space Ω = {t t ≥ 0} (continuous non - negative real numbers, because time can be any non - negative number).
Mechanical axis error: The sample points are x mm (x is a real number, for example, +0.01mm means 0.01mm larger than the design value and -0.02mm means 0.02mm smaller than the design value). The sample space Ω = {x x ∈ R} (a continuous set of real numbers, because the error can be positive, negative or zero).
The value of the sample space lies in transforming the abstract "random phenomenon" into a quantifiable mathematical object. For example, the sample space of coin - tossing consists of two points, and we can calculate the probability of "heads up". The sample space of the TV failure time is continuous time, and we can analyze the probability of "failure within 1 year of use". This is the foundation for subsequent probability and statistics.