Chapter 1: Q34E (page 50)

On page 38, we claimed that since about a $\frac{1}{n}$ fraction of n-bit numbers are prime, on average it is sufficient to draw $O (n)$ random n -bit numbers before hitting a prime. We now justify this rigorously. Suppose a particular coin has a probability p of coming up heads. How many times must you toss it, on average, before it comes up heads? (Hint: Method 1: start by showing that the correct expression is $\sum_{i = 1}^{\infty} i (1 - p)^{i - 1} p$ . Method 2: if E is the average number of coin tosses, show that $E = 1 + (1 - p) E$ ).

Short Answer

Expert verified

$\frac{1}{p}$ times the coin must be tossed on average before it comes up heads.

Step by step solution

Explain the given information

Consider that,P is the Probability of a coin to show up heads. Let,X be the number of times the coin should be tossed before outcome is head. The following equation is used to find the average number of coin tossed:

$E (X) = \sum_{I = 1}^{\infty} i \times P (i)$

Calculate, how may time the coin must be tossed on average, before it comes up heads.

Consider that $(i - 1)$ times the coins were thrown , and the $i$ th throwis a head. The probability of this event occurring is ,

$p (i) = {(1 - p)}^{i - 1} p$

The number of time the coin must be tossed on average can be calculated as follows,

$E = \sum_{i = 1}^{\infty} i \times P (I) E = \sum_{i = 1}^{\infty} i {(1 - p)}^{i - 1} p (1 - p) E = \sum_{i = 1}^{\infty} i {(1 - p)}^{i} p$

Simplify further,

localid="1659237715633" $E - (1 - p) E = p + \sum_{i = 1}^{\infty} {(1 - p)}^{i} p = p + \frac{(1 - p) p}{p} = 1$

Find as follows,

$\begin{array}{l} E - (1 - p) E = 1 \\ E - E + pE = 1 \\ pE = 1 \\ E = \frac{1}{p} \end{array}$

Therefore, $\frac{1}{p}$ times the coin must be tossed on average before it comes up heads.

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 Computer Science Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Explain the given information

Calculate, how may time the coin must be tossed on average, before it comes up heads.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Theory of Computation

Functional Programming

Computer Programming

Computer Network

Databases

Data Representation in Computer Science

Study anywhere. Anytime. Across all devices.