Q. 5.11 A point is chosen at random on a... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q. 5.11 (page 212)

A point is chosen at random on a line segment of
length $L$ . Interpret this statement, and find the probability
that the ratio of the shorter to the longer segment is
less than $\frac{1}{4}$ .

Short Answer

Expert verified

The required probability is $\frac{2}{5}$ .

Step by step solution

Step 1. Given Information.

Here, it is given that the length of a line segment is $L$ on which a point is chosen at random.

Step 2. Find the ratio of shorter to longer segment.

Let the point chosen at random on the given line segment be $X$ , which is uniformly distributed over the interval $(0, L)$ .

The probability density function of $X$ is given by:

$f (x) = \{\begin{cases} \frac{1}{L}, 0 鈮� x 鈮� L \end{cases}$

Ratio of shorter segment to longer segment will be the minimum of the following:

$\frac{X}{L - X}, \frac{L - X}{X}$

Step 3. Find the required probability.

The required probability will be:

$P \{M i n (\frac{X}{L - X}, \frac{L - X}{X}) < \frac{1}{4}\} = 1 - P \{M i n (\frac{X}{L - X}, \frac{L - X}{X}) > \frac{1}{4}\}$

Since, minimum of the two is greater than $\frac{1}{4}$ .

$1 - P (\frac{X}{L - X} > \frac{1}{4}, \frac{L - X}{X} > \frac{1}{4}) = 1 - P \{4 X > L - X, 4 (L - X) > X\} = 1 - P \{5 X > L, 4 L > 5 X\} = 1 - P \{X > \frac{L}{5}, X < \frac{4 L}{5}\} = 1 - P \{\frac{L}{5} < X < \frac{4 L}{5}\} = 1 - {鈭�}_{\frac{L}{5}}^{\frac{4 L}{5}} \frac{1}{L} d x = 1 - \frac{1}{L} (\frac{4 L}{5} - \frac{L}{5}) = 1 - \frac{3}{5} = \frac{2}{5}$

Therefore, the required probability is $\frac{2}{5} .$

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91影视

A point is chosen at random on a line segment of
length $L$ . Interpret this statement, and find the probability
that the ratio of the shorter to the longer segment is
less than $\frac{1}{4}$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the ratio of shorter to longer segment.

Step 3. Find the required probability.

One App. One Place for Learning.

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91影视

A point is chosen at random on a line segment oflength L. Interpret this statement, and find the probabilitythat the ratio of the shorter to the longer segment isless than 14.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the ratio of shorter to longer segment.

Step 3. Find the required probability.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Mechanics Maths

Geometry

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

A point is chosen at random on a line segment of
length $L$ . Interpret this statement, and find the probability
that the ratio of the shorter to the longer segment is
less than $\frac{1}{4}$ .