Suppose you drop a dye at random on the rectangular region shown in the figure. What is the probability that it will land inside the circle with a diameter of 1 m?
Step-by-step solution
To find: P(the drop lands inside the circle)
Idea: When a point is chosen at random in a region, every part of equal area is equally likely to receive it. So the probability is the fraction of the area that is favourable: area of circlearea of rectangle.
- Area of the rectangle = 3 × 2 = 6 m2.½ mark
- Radius of the circle = 12 × 1 = 0.5 m. Area of the circle = πr2 = π × (0.5)2 = π4 m2 ≈ 3.14 × 0.25 = 0.785 m2.1 mark
- P(inside the circle) = Area of circleArea of rectangle = π/46 = π24.1 mark
- = 0.7856 ≈ 0.13, about 13%. (With π = 227: 1184 ≈ 0.13.)½ mark
Check: The circle is clearly a small part of the rectangle (0.785 of 6 m2), so a probability a little above 18 = 0.125 is sensible ✓.
Answer to write in the exam
Area of rectangle = 3 × 2 = 6 m2
Radius of circle = 0.5 m
Area of circle = πr2 = π × (0.5)2 = π4 m2 ≈ 0.785 m2
P = Area of circleArea of rectangle = π/46 = π24
∴ P(lands inside the circle) = π24 ≈ 0.13
Common mistakes that cost marks
- Using the diameter as the radius: π × 12 = π gives π6 ≈ 0.52, four times too big. The radius is 0.5 m.
- Using the perimeter of the rectangle (10 m) instead of its area (6 m2).
- Comparing lengths (1 m out of 3 m) instead of areas. A drop lands on a region, so areas are compared.
How this can come in the exam
A point is chosen at random inside a square of side 4 cm. A smaller square of side 2 cm is drawn inside it. The probability that the point lies inside the smaller square is
- 12
- 14
- 18
- 116
Show answer
(B) 14
2 × 24 × 4 = 416 = 14.
A school garden is a rectangle 20 m by 14 m. In it there is a circular fountain of radius 7 m. A ball thrown from a window lands at a random point in the garden. (Use π = 227.)
(a) Find the area of the garden. (1)
(b) Find the area of the fountain. (1)
(c) Find the probability that the ball lands in the fountain, and the probability that it does not. (2)
Show answer
(a) 20 × 14 = 280 m2.(b) 227 × 7 × 7 = 154 m2.
(c) P(fountain) = 154280 = 1120 = 0.55. P(not) = 1 − 0.55 = 0.45.
Try one yourself
A dart lands at a random point of a square board of side 10 cm. A circle of radius 2 cm is painted on it. Find the probability that the dart lands inside the circle (use π = 3.14).
Show answer
3.14 × 4100 = 12.56100 = 0.1256 ≈ 0.13.
More questions like this
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