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Chapter 5: Problem 6

Let the sample space be \(S=\\{1,2,3,4,5,6,7,8,9,10\\} .\) Suppose that the outcomes are equally likely. Compute the probability of the event: \(F="\) a number divisible by three."

Short Answer

Expert verified
The probability is \( \frac{3}{10} \).

Step by step solution

01

Identify the Sample Space

The sample space, S, is given as the set of all possible outcomes: \[ S = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \]
02

Define the Event

The event F is defined as selecting a number that is divisible by three.
03

Find the Favorable Outcomes

Among the numbers in the sample space, identify which ones are divisible by three: \[ 3, 6, 9 \]
04

Count the Number of Favorable Outcomes

There are 3 numbers in the sample space that are divisible by three: 3, 6, and 9. So, the number of favorable outcomes is 3.
05

Calculate the Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes in the sample space. \[ P(F) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{10} \]

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Key Concepts

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

Sample Space
The concept of a sample space is fundamental in probability. It is the set of all possible outcomes of an experiment. In our exercise, the sample space, denoted by \(\text{S}\), is given as \[ S = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \]. This means that when an event occurs, it can only result in one of these 10 outcomes. Understanding the sample space helps us see all the possible results and ensures that we are considering the entire range of potential outcomes.
Favorable Outcomes
Favorable outcomes are the specific outcomes within our sample space that satisfy the condition of the event we are interested in. In the provided exercise, the event \(\text{F}\) is described as selecting a number that is divisible by three. To identify the favorable outcomes, we examine our sample space \( S = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \) and pick out the numbers divisible by three. These numbers are 3, 6, and 9. These are our favorable outcomes because they meet the criteria set by the event. The accuracy of our probability calculations depends on correctly identifying these outcomes.
Divisibility
Divisibility is a mathematical concept where one number can be divided by another without leaving a remainder. In simple terms, a number \( \text{a} \) is divisible by another number \( \text{b} \) if, after dividing \( \text{a} \) by \( \text{b} \), the result is an integer, and the remainder is zero. For instance, in the exercise, numbers like 3, 6, and 9 are divisible by 3 because:
  • \( \frac{3}{3} = 1 \) (no remainder)
  • \( \frac{6}{3} = 2 \) (no remainder)
  • \( \frac{9}{3} = 3 \) (no remainder)
Divisibility rules help in effortlessly determining if one number can be evenly divided by another, which is crucial for identifying favorable outcomes in probability exercises.

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Most popular questions from this chapter

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You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?
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