Chapter 3: Q.3.88 (page 106)

In Example $5$ e, what is the conditional probability that the ith coin was selected given that the ﬁrst n trials all result in heads?

Short Answer

Expert verified

$P (C_{i} ∣ F_{n}) = \frac{i^{n}}{\sum_{j = 0}^{k} j^{n}}, the beginning formula is given in the example.$

The denominator is the power sum, for which there are no explicit formulae.

Step by step solution

Given Information

The conditional probability that the ith coin was selected given that the ﬁrst n trials all result in heads

Given:

Example $5$ _e

Events F_n,C_i

i= $0, 1, . . . K$

Explanation

Probabilities:

$P (C_{i}) = \frac{1}{k + 1}, i = 0, 1, \dots, k$

$P (H ∣ C_{i}) = \frac{i}{k}$

Events of flipping heads in different flips are independent given C_i, Therefore:

$P (F_{n} ∣ C_{i}) = \prod_{i = 1}^{n} P (H ∣ C_{i}) = {(\frac{i}{k})}^{n}$

$Calculate P (C_{i} ∣ F_{n})$

From the example

$P (C_{i} ∣ F_{n}) = \frac{P (C_{i} F_{n})}{P (F_{n})} = \frac{P (F_{n} ∣ C_{i}) P (C_{i})}{\sum_{j = 0}^{k} P (F_{n} ∣ C_{j}) P (C_{j})} = \frac{{(\frac{i}{k})}^{n} \cdot \frac{1}{k + 1}}{\sum_{j = 0}^{k} {(\frac{j}{k})}^{n} \cdot \frac{1}{k + 1}}$

$Reducing the fraction by factor \frac{1}{k^{n}} \cdot \frac{1}{k + 1}$

$P (C_{i} ∣ F_{n}) = \frac{i^{n}}{\sum_{j = 0}^{k} j^{n}}$

The number below is a power sum, and it cant be calculated by a simple explicit formula, some of the variations.

$S_{n} (k) = \sum_{j = 0}^{k} j^{n} = ζ (- n) - ζ (- n, k + 1)$

$Where ζ is the Reimann zeta function, f (n, a)$

Final Answer

$P (C_{i} ∣ F_{n}) = \frac{i^{n}}{\sum_{j = 0}^{k} j^{n}}, the beginning formula is given in the example.$

The denominator is the power sum, for which there are no explicit formulae.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Example $5$ e, what is the conditional probability that the ith coin was selected given that the ﬁrst n trials all result in heads?

Short Answer

Step by step solution

Given Information

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Mechanics Maths

Calculus

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Example 5e, what is the conditional probability that the ith coin was selected given that the ﬁrst n trials all result in heads?

Short Answer

Step by step solution

Given Information

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Mechanics Maths

Calculus

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Example $5$ e, what is the conditional probability that the ith coin was selected given that the ﬁrst n trials all result in heads?