6. Find the probability that a point chosen at random inside the larger circle shown here will also fall inside the smaller circle. 3 cm 4 cm
6. Find the probability that a point chosen at random inside the larger circle shown here will also fall inside the smaller circle. 3 cm 4 cm
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem 6: Probability of Random Point in a Circle**
6. Find the probability that a point chosen at random inside the larger circle shown here will also fall inside the smaller circle.
The diagram shows two concentric circles. The larger circle has a radius of 4 cm, and the smaller circle has a radius of 3 cm.
---
### Explanation
To find the probability that a point chosen at random inside the larger circle will also fall inside the smaller circle, follow these steps:
1. **Calculate the area of both circles:**
- The area \(A\) of a circle is given by the formula: \(A = \pi r^2\).
- Area of the smaller circle (radius \(r = 3\) cm):
\[
A_{\text{small}} = \pi (3)^2 = 9\pi \, \text{cm}^2
\]
- Area of the larger circle (radius \(r = 4\) cm):
\[
A_{\text{large}} = \pi (4)^2 = 16\pi \, \text{cm}^2
\]
2. **Find the probability:**
- The probability \(P\) that a point randomly chosen in the larger circle will be in the smaller circle is the ratio of the areas of the smaller circle to the larger circle:
\[
P = \frac{A_{\text{small}}}{A_{\text{large}}} = \frac{9\pi}{16\pi} = \frac{9}{16}
\]
Therefore, the probability that a point chosen at random inside the larger circle will also fall inside the smaller circle is \(\frac{9}{16}\).
### Diagram Description:
The diagram accompanying this problem consists of two concentric circles. The smaller circle has a radius labeled as 3 cm, and the larger circle has a radius labeled as 4 cm, both measured from the center of the circles to their respective circumferences.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd89c40f4-4529-4a29-9a11-662a2483fc15%2Fa8e50563-747c-4387-a952-00ec99d524dc%2Fwqguqme_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Probability of Random Point in a Circle**
6. Find the probability that a point chosen at random inside the larger circle shown here will also fall inside the smaller circle.
The diagram shows two concentric circles. The larger circle has a radius of 4 cm, and the smaller circle has a radius of 3 cm.
---
### Explanation
To find the probability that a point chosen at random inside the larger circle will also fall inside the smaller circle, follow these steps:
1. **Calculate the area of both circles:**
- The area \(A\) of a circle is given by the formula: \(A = \pi r^2\).
- Area of the smaller circle (radius \(r = 3\) cm):
\[
A_{\text{small}} = \pi (3)^2 = 9\pi \, \text{cm}^2
\]
- Area of the larger circle (radius \(r = 4\) cm):
\[
A_{\text{large}} = \pi (4)^2 = 16\pi \, \text{cm}^2
\]
2. **Find the probability:**
- The probability \(P\) that a point randomly chosen in the larger circle will be in the smaller circle is the ratio of the areas of the smaller circle to the larger circle:
\[
P = \frac{A_{\text{small}}}{A_{\text{large}}} = \frac{9\pi}{16\pi} = \frac{9}{16}
\]
Therefore, the probability that a point chosen at random inside the larger circle will also fall inside the smaller circle is \(\frac{9}{16}\).
### Diagram Description:
The diagram accompanying this problem consists of two concentric circles. The smaller circle has a radius labeled as 3 cm, and the larger circle has a radius labeled as 4 cm, both measured from the center of the circles to their respective circumferences.
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