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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 6: Q.6.4 (page 275)

Solve Buffon’s needle problem when L > D. answer: 2L πD(1 − sin θ) + 2θ/π, where cos θ = D/L.

Short Answer

Expert verified

The Buffon's needle problem is proved where L > D.

Step by step solution

01

Content Introduction

Given a floor with evenly spaced parallel lines a distance d apart, Buffon's needle problem asks for the probability that a needle of length l would land on a line.

02

Content Explanation

When L > D, needle will surely cut at least one line for all $θ$ such that $L \cos (θ) \geq D$

Therefore from $0 t o θ$ (such that $\cos {(θ)}^{1} = \frac{D}{L}$ needle will surely cut one line at least. Also, will $θ$ be on both sides.

Therefore, out of $π$ angle available $2 θ$ will give a sure event for angles more than $θ$ .

03

Explanation Proof

$P {X < \frac{L}{2} \cos (θ) = \iint f_{X} (x) f_{0} (y) d x d y = \frac{4}{πD} \int_{θ^{1}}^{\frac{π}{2}} \int_{0}^{\frac{L}{2} \cos (y)} d x d y = \frac{4}{πD} \int_{θ}^{\frac{π}{2}} \frac{L}{2} \cos (y) d y \frac{2 L}{πD} (1 - \sin (θ^{1})$

Total probability $\frac{2 L}{πD} (1 - \sin {(θ)}^{1} + \frac{2 θ}{π}$

where $\cos {(θ)}^{1} = \frac{D}{L}$

Hence proved.

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Most popular questions from this chapter

Each throw of an unfair die lands on each of the odd numbers $1, 3, 5$ with probability C and on each of the even numbers with probability $2 C$ .

(a) Find C.

(b) Suppose that the die is tossed. Let X equal $1$ if the result is an even number, and let it be $0$ otherwise. Also, let Y equal $1$ if the result is a number greater than three and let it be $0$ otherwise. Find the joint probability mass function of X and Y. Suppose now that $12$ independent tosses of the die are made.

(c) Find the probability that each of the six outcomes occurs exactly twice.

(d) Find the probability that $4$ of the outcomes are either one or two, $4$ are either three or four, and $4$ are either five or six.

(e) Find the probability that at least $8$ of the tosses land on even numbers.

Verify Equation $(1.2)$ . $P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$

Compute the density of the range of a sample of size $n$ from a continuous distribution having density function $f$ .

Consider independent trials, each of which results in outcome i, i = 0, 1, ... , k, with probability pi, k i=0 pi = 1. Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.

(a) Find P{N = n}, n Ú 1.

(b) Find P{X = j}, j = 1, ... , k.

(c) Show that P{N = n, X = j} = P{N = n}P{X = j}.

(d) Is it intuitive to you that N is independent of X?

(e) Is it intuitive to you that X is independent of N?

Consider a sample of size $5$ from a uniform distribution over $(0, 1)$ . Compute the probability that the median is in the interval $(\frac{1}{4}, \frac{3}{4})$ .

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