Spinning Arrow and Geometric Probability | Class 10 Maths

Quick answer: For an arrow that is equally likely to stop at any of the numbers $1$ to $8$: $P(8) = \frac{1}{8}$, $P(\text{odd number}) = \frac{4}{8} = \frac{1}{2}$, $P(\text{number greater than } 2) = \frac{6}{8} = \frac{3}{4}$, and $P(\text{number less than } 9) = \frac{8}{8} = 1$ (a sure event). When an object lands at random on a region, probability can also be found using areas: $P = \frac{\text{favourable area}}{\text{total area}}$; for a die dropped on a $3\text{ m} \times 2\text{ m}$ rectangle containing a circle of diameter $1\text{ m}$, $P = \frac{\pi}{24} \approx 0.13$.

Most probability questions count outcomes: the faces of a die, the cards in a pack, or the numbers on a spinner. This guide solves the NCERT spinning arrow problem, and then looks at geometric probability, where probability is found by comparing areas instead of counting.

In CBSE Class 10 Mathematics, Chapter 14 (Probability), the spinning arrow problem is a well-known NCERT question.

Syllabus note: The spinning arrow problem is part of the NCERT exercise. The geometric (area) probability problem was an optional NCERT exercise and is not part of the current rationalised textbook. It is included here as enrichment.

What You Will Learn

• How to find probabilities for a spinner with equally likely numbers
• What a sure event is
• How to find probability using areas
• A solved example: a die dropped on a rectangle containing a circle

1. Problem 1: The Spinning Arrow (NCERT)

Question

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers $1, 2, 3, 4, 5, 6, 7, 8$, and these are equally likely outcomes. What is the probability that it will point at:

(i) $8$?

(ii) an odd number?

(iii) a number greater than $2$?

(iv) a number less than $9$?

Solution

There are $8$ equally likely outcomes: $1, 2, 3, 4, 5, 6, 7, 8$.

(i) Pointing at 8. Only $1$ favourable outcome:

$P(8) = \frac{1}{8}$

(ii) An odd number. The odd numbers are $1, 3, 5, 7$, which is $4$ outcomes:

$P(\text{odd}) = \frac{4}{8} = \frac{1}{2}$

(iii) A number greater than 2. These are $3, 4, 5, 6, 7, 8$, which is $6$ outcomes:

$P(\text{greater than } 2) = \frac{6}{8} = \frac{3}{4}$

(iv) A number less than 9. All $8$ numbers are less than $9$:

$P(\text{less than } 9) = \frac{8}{8} = 1$

Key point: An event that is certain to happen, such as getting a number less than $9$ here, is called a sure event. Its probability is $1$.

2. Geometric Probability: Using Areas (Enrichment)

When an object lands at random on a region, every point of the region is equally likely. Then:

$P(\text{landing in a part of the region}) = \frac{\text{area of that part}}{\text{total area of the region}}$

3. Problem 2: A Die Dropped on a Rectangle

Question

A die is dropped at random on a rectangular region measuring $3\text{ m} \times 2\text{ m}$. Inside it is a circle of diameter $1\text{ m}$. What is the probability that the die lands inside the circle?

Solution

Step 1: Total area.

$\text{Area of rectangle} = 3 \times 2 = 6\text{ m}^2$

Step 2: Area of the circle. The diameter is $1\text{ m}$, so the radius is $\frac{1}{2}\text{ m}$:

$\text{Area of circle} = \pi \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}\text{ m}^2$

Step 3: Probability.

$P = \frac{\pi/4}{6} = \frac{\pi}{24} \approx 0.13$

Answer: The probability is $\frac{\pi}{24}$, about $0.13$.

4. Summary Table

5. Exam Tips and Common Mistakes

Exam Tip: List the favourable outcomes before writing each probability, e.g. "odd numbers: 1, 3, 5, 7".

Exam Tip: It is fine to leave an answer in terms of $\pi$, such as $\frac{\pi}{24}$, unless a decimal is asked for.

Common Mistake: Using the diameter as the radius. For a circle of diameter $1\text{ m}$, $r = 0.5\text{ m}$. Using $r = 1\text{ m}$ gives an area four times too large.

Common Mistake: Counting $2$ as "greater than $2$". It is not; the numbers greater than $2$ are $3$ to $8$.

Frequently Asked Questions

What is the probability that the spinning arrow points at an odd number?

There are 4 odd numbers out of 8, so the probability is 1/2.

What is the probability that the arrow points at a number less than 9?

All 8 numbers are less than 9, so the probability is 1. This is a sure event.

What is a sure event?

An event that is certain to happen. Its probability is 1.

How do you find probability using areas?

Divide the area of the favourable region by the total area of the region where the object can land.

What is the probability that a die dropped on a 3 m × 2 m rectangle lands in a circle of diameter 1 m?

π/24, which is about 0.13.