Introduction

Today we’re going to learn about a cool topic in maths called ‘Geometric Probability’. Now, this may sound intimidating, but don’t worry! We’re going to break it down into bite-sized pieces so that it’s easy to understand.

What is Geometric Probability?

Geometric Probability is a method of probability analysis that uses geometric principles. In other words, we use shapes and their measurements to figure out the likelihood of an event happening.

Here’s an example:

Let’s say you’re playing darts and you want to hit the bullseye. The dartboard is a circle and the bullseye is a smaller circle in the middle. The probability of hitting the bullseye is the area of the bullseye divided by the total area of the dartboard.

Key Rules

  1. Area and Length: We use the total area (for 2D shapes) or total length (for 1D shapes) to represent all possible outcomes.
  2. Favourable Outcomes: The area or length representing favourable outcomes is divided by the total area or length.
  3. Probability Range: The calculated probability will always be between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

Tips and Tricks

  1. Draw a Diagram: It can be helpful to draw a diagram to visualize the problem.
  2. Formulate the Problem: Express the problem as a ratio of two areas or lengths.
  3. Be Consistent with Units: Make sure your measurements are in the same units before calculating.

Practice Questions

Easy Level

  1. A square has a side length of 10 cm. What is the probability that a randomly thrown dart lands in a 2 cm by 2 cm square in the corner of the larger square?
  2. A rectangle has dimensions 20 cm by 10 cm. What is the probability that a randomly thrown dart lands in a 5 cm by 5 cm square in the corner of the rectangle?
  3. A circle has a radius of 10 cm. What is the probability that a randomly thrown dart lands in a smaller circle within the larger circle that has a radius of 2 cm?
  4. A straight line segment has a length of 15 cm. What is the probability that a randomly placed point lies within the first 3 cm of the line?
  5. A triangle has an area of 50 cm². What is the probability that a randomly thrown dart lands in a smaller triangle within the larger triangle that has an area of 10 cm²?

Medium Level

  1. A square has a side length of 8 m. What is the probability that a randomly thrown dart lands in a smaller square within the larger square that has a side length of 3 m?
  2. A rectangle has dimensions 25 m by 15 m. What is the probability that a randomly thrown dart lands in a smaller rectangle within the larger rectangle that has dimensions 5 m by 3 m?
  3. A circle has a radius of 12 m. What is the probability that a randomly thrown dart lands in a smaller circle within the larger circle that has a radius of 3 m?
  4. A straight line segment has a length of 20 m. What is the probability that a randomly placed point lies within the first 5 m of the line?
  5. A triangle has an area of 100 m². What is the probability that a randomly thrown dart lands in a smaller triangle within the larger triangle that has an area of 25 m²?

Hard Level

  1. An equilateral triangle has a side length of 12 cm. What is the probability that a randomly thrown dart lands in a smaller equilateral triangle within the larger triangle that has a side length of 3 cm?
  2. A rectangle has dimensions 30 cm by 20 cm. What is the probability that a randomly thrown dart lands in a smaller rectangle within the larger rectangle that has dimensions 8 cm by 5 cm?
  3. A circle has a radius of 15 cm. What is the probability that a randomly thrown dart lands in a smaller circle within the larger circle that has a radius of 4 cm?
  4. A straight line segment has a length of 25 cm. What is the probability that a randomly placed point lies within the first 7 cm of the line?
  5. A pentagon has an area of 200 cm². What is the probability that a randomly thrown dart lands in a smaller pentagon within the larger pentagon that has an area of 50 cm²?
 The answers and explanations are as follows:

Answers

Easy Level

  1. The probability is 0.04. The total area of the larger square is 100 cm² and the area of the smaller square is 4 cm². So, the probability is 4/100 = 0.04.
  2. The probability is 0.025. The total area of the rectangle is 200 cm² and the area of the smaller square is 5 cm². So, the probability is 5/200 = 0.025.
  3. The probability is 0.04. The total area of the larger circle is 314.16 cm² and the area of the smaller circle is 12.57 cm². So, the probability is 12.57/314.16 = 0.04.
  4. The probability is 0.2. The total length of the line segment is 15 cm and the length of the smaller segment is 3 cm. So, the probability is 3/15 = 0.2.
  5. The probability is 0.2. The total area of the larger triangle is 50 cm² and the area of the smaller triangle is 10 cm². So, the probability is 10/50 = 0.2.

Medium Level

Answers and explanations are similar to the easy level but with different measurements.

Hard Level

Answers and explanations are similar to the easy and medium levels but with different shapes and measurements.