【People's Education Press】Grade 9 Math, First Semester: The Coincidence and Certainty in Life: Categorization and Analysis of Events

Welcome to the world of probability! We once lived in the certainty of geometry: given a fixed radius $r$ and distance $d$, the position of a point on a circle is uniquely determined. Yet in reality, when we roll a die or draw a lottery slip, outcomes are often filled with 'chance'. In this lesson, we’ll learn how to classify these phenomena using mathematical language.

From Determinism to Randomness

In mathematics, based on the likelihood of an event occurring under specific conditions, we can categorize events into three main types:

1. Certain Events

Under certain conditions,will inevitably occuran event that will definitely happen. For example: in the same circle, a diameter perpendicular to a chord bisects it. When the condition (perpendicular and passing through the center) is met, the result (bisecting) occurs with 100% certainty.

2. Impossible Events

Under certain conditions,will never occuran event that will never happen. For example: under the inscribed angle theorem, the inscribed angle subtended by the same arc is greater than the central angle. The probability of such an event is 0.

3. Random Events

Under certain conditions,may or may not occuran event that may or may not happen. For example: rolling a six on a die. Before the action takes place, we cannot predict the exact outcome.

Geometric Symmetry and Probability Balance

The axial, central, and rotational symmetry of circles (key concept: circle symmetry) symbolize an ideal state of balance. This aligns logically with the assumption of 'uniformity' in random experiments within probability theory. When we say a die is fair, we're essentially assuming its physical symmetry leads to balanced probabilities for all outcomes.

🎯 Core Mental Model

The key to identifying an event type lies in whether, underspecific conditions, the outcome is 'uniquely determined' or 'multiple possibilities exist'.

QUESTION 1

The radius of ⊙O is 10 cm. Based on the following distances from point P to center O, determine the positional relationship between point P and ⊙O: (1) 8 cm; (2) 10 cm; (3) 12 cm.

(1) Inside the circle; (2) On the circle; (3) Outside the circle

(1) On the circle; (2) Inside the circle; (3) Outside the circle

(1) Outside the circle; (2) On the circle; (3) Inside the circle

(1) Inside the circle; (2) Outside the circle; (3) On the circle

QUESTION 2

Identify which of the following events are certain, impossible, or random: (1) Water typically boils at 100°C; (2) A basketball player misses a free throw shot; (3) Rolling a six on a die; (4) Drawing any triangle with interior angles summing to 360°; (5) Encountering a red light at a traffic intersection; (6) A shooter hitting the bullseye on one shot.

Certain: (1); Impossible: (4); Random: (2)(3)(5)(6)

Certain: (1)(4); Impossible: None; Random: (2)(3)(5)(6)

Certain: (1); Impossible: (2)(4); Random: (3)(5)(6)

Certain: (1); Impossible: (4)(5); Random: (2)(3)(6)

QUESTION 3

Given that the ratio of land area to ocean area on Earth's surface is approximately 3:7, if a meteorite lands on Earth, which is more likely: landing on land or landing in the ocean?

Landing in the ocean is more likely

Landing on land is more likely

Both have equal likelihood

Cannot be determined

QUESTION 4

The table below records a player's shooting results: number of attempts n (50, 100, 150, 200, 250, 300, 500); number of successful shots m (28, 60, 78, 104, 123, 152, 251). What is the player's approximate shooting probability?

0.5

0.6

0.4

0.7

QUESTION 5

Which of the following statements about random events is correct?

The frequency of a random event's occurrence is its probability

Whether a random event occurs in a single trial is unpredictable

Certain events do not belong to deterministic events

The probability of an impossible event occurring is 1