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Geometric Probability: Choosing a Random Point

When a point is chosen completely at random from a figure, the chance that it lands in a smaller part is (size of the part) ÷ (size of the whole). Size means length for a segment or arc, and area for a flat figure.

🎬 Step-by-step story

  1. Here is a square board. A dart can land on any spot of it. All the spots together are the whole square.
  2. Now a blue circle appears. This is the target. We want the dart to land inside it.
  3. We throw 10 darts. Red darts hit the target, grey darts miss. Count the red ones: hits ÷ throws.
  4. Now 300 darts. The fraction of red darts settles near a fixed number. That number is blue area ÷ square area.
  5. Same idea on a bar. The bar has length 10 and the green part has length 4. Darts hit green about 4 out of 10 times.
  6. Your turn. Slide the radius to change the target, throw darts and compare the hits with the area ratio.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why do we use area and not count points?

The board has endless points. We cannot count them, so we compare sizes. Look at the blue area and the whole square in the 3D.

Why do the red darts not give exactly the area ratio with 10 darts?

Few throws are noisy. Luck matters a lot with 10 darts. Watch how the number becomes steady with 300.

What happens when I throw many more darts?

The hit fraction gets closer to the area ratio. This is the law of large numbers.

Is a segment problem the same idea?

Yes. Only the measure changes: length instead of area. The green part is 4 of 10.

What if I make the target bigger?

More area gives a higher probability. Move the radius slider and watch both numbers rise together.

What is geometric probability?

In school probability we often count outcomes: 6 faces of a die. But some experiments have endless outcomes. A point on a board has endless possible places. We cannot count them.

So we measure them instead. If every place is equally likely, then

P = size of the favourable part ÷ size of the whole figure

"Size" means the same kind of measure for both: length, area (or, later, volume).

A random point in a flat figure (area)

Pick a point at random inside a square of area 64. A circle of area 20 sits inside it. The chance of landing in the circle is 20 ÷ 64 = 0.31.

For a circle of radius r in a square of side a: P = πr² ÷ a². The bigger the target, the bigger the chance. If the target covers the whole figure, P = 1. If it has no area, P = 0.

A point exactly on a boundary line has zero area, so the chance of hitting one exact point is 0.

A random point on a segment (length)

A segment is a straight piece of line. Pick a point on a segment of length 10. A part of length 4 is green. Then P(green) = 4 ÷ 10 = 0.4.

Time works like a segment. A bus comes every 20 minutes. If you arrive at a random moment, the chance you wait less than 5 minutes is 5 ÷ 20 = 0.25.

A random point on an arc (angle)

An arc is a curved piece of a circle. A spinner stops at a random point of the circle edge. Arc length is proportional to its angle. So P = arc angle ÷ 360°.

A sector with angle 90° gives P = 90 ÷ 360 = 0.25. Use angle or arc length, whichever you are given. Just do not mix them.

Steps to solve any problem

  1. Draw the whole figure and the favourable part.
  2. Decide the measure: length, area or angle.
  3. Find the size of the part and the size of the whole.
  4. Divide: part ÷ whole.
  5. Check that the answer is between 0 and 1.

Tip: sometimes it is easier to find the part that does not count. Then P = 1 − P(not).

Try it

At home, draw a 20 cm square and a circle of radius 6 cm inside it. Drop 30 dry beans from above with your eyes closed. Count beans in the circle. Predict first: π × 36 ÷ 400 = 0.28. Is your count close? In the 3D, throw 100 darts and change the radius slider.

Key formulas and definitions

Worked examples

1. A square board has side 10 cm. A circle of radius 3 cm is drawn inside. A dart lands at random. Find the chance it hits the circle.

Whole area = 10 × 10 = 100. Circle area = π × 3² = 28.3. P = 28.3 ÷ 100 = 0.283.

2. A bus comes every 20 minutes. You arrive at a random time. What is the chance you wait less than 5 minutes?

Think of 20 minutes as a segment. Favourable part = 5 minutes. P = 5 ÷ 20 = 0.25.

3. A spinner has a red sector of 90°. Find P(it stops on red).

P = 90 ÷ 360 = 1/4 = 0.25.

4. A point is picked at random in a rectangle 3 by 4. What is the chance it lies in the lower right triangle cut by a diagonal?

Rectangle area = 12. The triangle is half of it = 6. P = 6 ÷ 12 = 0.5.

5. A square has side 10. What is the chance a random point is more than 2 units away from every side?

Such points form an inner square of side 10 − 4 = 6. Area 36. P = 36 ÷ 100 = 0.36.

6. A target is a circle of radius 5. The red ring lies between radius 3 and radius 5. Find P(red ring).

Ring area = π(25 − 9) = 16π. Whole = 25π. P = 16 ÷ 25 = 0.64.

7. Two friends each arrive at a random time in one hour and wait 15 minutes for the other. What is the chance they meet?

Draw a 60 by 60 square (arrival times). They meet if the times differ by at most 15. The "do not meet" part is two triangles with legs 45: total area 45 × 45 = 2025. Whole = 3600. P(not) = 0.5625, so P(meet) = 1 − 0.5625 = 0.4375.

Common mistakes

Practice quiz

1. The chance of a random point landing in part A of a figure is:
2. A random point is picked on a segment of length 10. A part has length 2. P =
3. A spinner sector of 180° has probability:
4. What is the chance of hitting one exact point on a board?
5. If the part is the whole figure, P =

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is geometric probability in simple words?

It is the chance that a point picked at random lands in a chosen part of a shape. You find it by dividing the size of the part by the size of the whole.

Why can we not just count outcomes?

A shape has endless points, so there is nothing to count. We use length, area or angle instead.

Does the answer depend on where the target is placed?

No. If every point is equally likely, only the size of the target matters, not its position, as long as it is inside the figure.

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