/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Express all probabilities as fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 31

Express all probabilities as fractions. The International Morse code is a way of transmitting coded text by using sequences of on \(/\) off tones. Each character is 1 or 2 or 3 or 4 or 5 segments long, and each segment is either a dot or a dash. For example, the letter \(G\) is transmitted as two dashes followed by a dot, as in \(--\bullet\). How many different characters are possible with this scheme? Are there enough characters for the alphabet and numbers?

Short Answer

Expert verified
There are 62 different characters possible. Yes, there are enough characters for the alphabet and numbers.

Step by step solution

01

Understand Segment Lengths

Characters can be composed of 1 to 5 segments, where each segment is either a dot or a dash. Thus, each segment has 2 possible states.
02

Calculate 1-Segment Characters

For 1-segment characters, there are 2 possibilities: a dot or a dash. Thus, the total is: \[ 2^1 = 2 \]
03

Calculate 2-Segment Characters

For 2-segment characters, each segment can be a dot or a dash. Thus, the total is: \[ 2^2 = 4 \]
04

Calculate 3-Segment Characters

For 3-segment characters, each segment can be a dot or a dash. Thus, the total is: \[ 2^3 = 8 \]
05

Calculate 4-Segment Characters

For 4-segment characters, each segment can be a dot or a dash. Thus, the total is: \[ 2^4 = 16 \]
06

Calculate 5-Segment Characters

For 5-segment characters, each segment can be a dot or a dash. Thus, the total is: \[ 2^5 = 32 \]
07

Calculate Total Number of Characters

Sum the possible characters for each segment length: \[ 2 + 4 + 8 + 16 + 32 = 62 \]
08

Compare with Required Characters

There are 26 letters in the alphabet and 10 numbers, totaling to 36 characters. Since 62 > 36, there are enough characters in the scheme.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Morse code
Morse code is a method of encoding text characters using sequences of dots and dashes. Each dot or dash represents a segment. Depending on the combination and length of these sequences, different characters are formed. For example, the letter G is represented by two dashes followed by a dot: --路. This system is used in telecommunication to transmit text over long distances, often by sound or light signals.
probability
Probability helps us to determine the likelihood of specific occurrences. In Morse code, each segment (dot or dash) has an equal probability of appearing. Calculating the number of possible characters involves determining the total combinations for 1 to 5 segments, each with 2 possible states (dot or dash). For instance, the probabilities for each scenario are:
  • For 1-segment: 2 possible characters (dot, dash)
  • For 2-segment: 2^2 = 4 possible characters
  • For 3-segment: 2^3 = 8 possible characters
  • For 4-segment: 2^4 = 16 possible characters
  • For 5-segment: 2^5 = 32 possible characters

Adding all possible characters gives us 62 different Morse code characters. Considering there are 26 letters in the alphabet and 10 numbers, a total of 36 characters is sufficient, making Morse code fit for representing all needed characters.

combinatorics
Combinatorics is a branch of mathematics focused on counting combinations and arrangements. In the case of Morse code, it helps to calculate the total number of unique characters possible with given segments. By understanding that each segment in Morse code can be either a dot or a dash, we use combinatorial methods to find the total number of possible combinations for sequence lengths from 1 to 5 segments.

Using the formula for combinations with replacement, we can calculate the total number of characters:
  • 1-segment: 2
  • 2-segment: 4
  • 3-segment: 8
  • 4-segment: 16
  • 5-segment: 32
Summing these values, we obtain 2 + 4 + 8 + 16 + 32 = 62. Combinatorics confirms that there are 62 unique Morse code characters possible with up to 5 segments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Composite Water Samples The Fairfield County Department of Public Health tests water for the presence of \(E\). coli (Escherichia coli) bacteria. To reduce laboratory costs, water samples from 10 public swimming areas are combined for one test, and further testing is done only if the combined sample tests positive. Based on past results, there is a \(0.005\) probability of finding \(E\). coli bacteria in a public swimming area. Find the probability that a combined sample from 10 public swimming areas will reveal the presence of \(E\). coli bacteria. Is that probability low enough so that further testing of the individual samples is rarely necessary?

Find the probability. Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)

Express all probabilities as fractions. A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?

Express all probabilities as fractions. A thief steals an ATM card and must randomly guess the correct pin code that consists of four digits (each 0 through 9 ) that must be entered in the correct order. Repetition of digits is allowed. What is the probability of a correct guess on the first try?

Answer the given questions that involve odds. When the horse California Chrome won the 140th Kentucky Derby, a \(\$ 2\) bet on a California Chrome win resulted in a winning ticket worth \(\$ 7\). a. How much net profit was made from a \(\$ 2\) win bet on California Chrome? b. What were the payoff odds against a California Chrome win? c. Based on preliminary wagering before the race, bettors collectively believed that California Chrome had a \(0.228\) probability of winning. Assuming that \(0.228\) was the true probability of a California Chrome victory, what were the actual odds against his winning? d. If the payoff odds were the actual odds found in part (c), what would be the worth of a \(\$ 2\) win ticket after the California Chrome win?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.