Geometric probability

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Target zones (example figures, made up for practice)

   zone	outer radius (cm)	score
   bullseye	3	10
   middle ring	8	5
   outer ring	12	1

2. 1.

   A beanbag lands at a random spot on Bo's target mat, shown on the card. What is the chance it scores at least 5 points?

   • 4/9
   • 55/144
   • 2/3
   • 5/9
3. 2.

   A bus comes at a random moment within a 30-minute window. Jun asks for the chance the wait is under 10 minutes. Which measure fits, and why?

   • Area: time passes in two directions at once, so compare the areas of the two time spans.
   • No measure works: a chance needs a list of equally likely outcomes to count them one by one in order.
   • Length: moments are spread along one line of 30 minutes, so use favorable length ÷ total length.
   • Angle: a clock face is round, so compare angles out of 360°.
4. 3.

   D and E are the midpoints of sides AB and AC of triangle ABC. A seed lands at a random spot in Kira's triangular garden ABC. What is the chance it lands in triangle ADE?

   • 3/4
   • 1/3
   • 1/4
   • 1/2
5. 4.

   Luz argues: the bullseye has radius 3 cm and the target mat radius 12 cm, so a random beanbag hits the bullseye with chance 3 ÷ 12 = 1/4. Where is the flaw?

   • The flaw is that the chance should add the two radii: 3/12 + 3/12.
   • There is no flaw, since radius ratios give the chances for round targets.
   • The flaw is the 12: the mat's radius should be 6, half of 12.
   • The chance follows area, not radius: 9π ÷ 144π = 1/16, so 1/4 is too big.
6. 5.

   A point is chosen at random in Milo's 10 cm by 10 cm square. Given that it is in the left half, what is the chance it is within 2 cm of the left edge?

   • 1/5
   • 2/5
   • 2/3
   • 3/5
7. 6.

   A circle of radius 4 cm is inscribed in Noah's square tile, touching all four sides. A random point is chosen on the tile. About how likely is it to land outside the circle?

   • 1 − π/4 ≈ 0.21
   • 1 − 2/π ≈ 0.36
   • π/4 ≈ 0.79
   • 1 − π/8 ≈ 0.61
8. 7.

   A point (x, y) is chosen at random in the square 0 ≤ x ≤ 6, 0 ≤ y ≤ 6, for Pia's grid game. What is the chance that x + y ≤ 3?

   • 1/2
   • 1/8
   • 7/8
   • 1/12
9. 8.

   Two friends arrive at random times between 1:00 and 2:00, each waiting 10 minutes. With x and y minutes after 1:00, which region of the 60 by 60 square means they meet, for Tala?

   • x + y ≤ 10: a small triangle at the corner of the square
   • x ≤ 10 and y ≤ 10: a small square at the corner of the square
   • x − y = 10: a single line across the square from edge to edge
   • |x − y| ≤ 10: a band along the diagonal
10. 9.

    Zee argues: a spinner with a 90° red sector lands in red with chance 1/4, so after one red the next spin lands in red with chance 1/16. What is the flaw?

    • The flaw is the 1/4: red should be 1/2, since it either lands in red or not.
    • No flaw: after one red, the spinner is less likely to land on red again, so 1/16 is right.
    • The flaw is that the next spin should be 1/4 + 1/4 = 1/2 after one red.
    • 1/16 is the chance of red twice; each spin is independent, so the next spin is still 1/4.
11. 10.

    Two numbers x and y are chosen at random from 0 to 1 for Bo's game. Which region is the event x + y < 1/2, and what is its chance?

    • The triangle under the line x + y = 1, with area 1/2, so the chance is 1/2.
    • A line, x + y = 1/2, so the chance is 0 since a line has no area.
    • The triangle under the line x + y = 1/2, with area 1/8, so the chance is 1/8.
    • The square with x and y each below 1/2, with an area of 1/4, so the chance is 1/4 as well.