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Chapter 4: Problem 14

A single die is rolled. What is the probability that the number on top is the following? a. \(A 3\) b. An odd number c. A number less than 5 d. A number no greater than 3

Short Answer

Expert verified
a. The probability of rolling a 3 is \(\frac{1}{6}\). b. The probability of rolling an odd number is \(\frac{1}{2}\). c. The probability of rolling a number less than 5 is \(\frac{2}{3}\). d. The probability of rolling a number no greater than 3 is \(\frac{1}{2}\).

Step by step solution

01

Probability of rolling a 3

The event is rolling a 3. There is exactly one outcome that fulfills this condition, which is rolling the number 3. So the probability is \(\frac{1}{6}\).
02

Probability of rolling an odd number

The event is rolling an odd number. There are three outcomes that fulfill this condition: rolling a 1, 3, or 5. So the probability is \(\frac{3}{6} = \frac{1}{2}\).
03

Probability of rolling a number less than 5

The event is rolling a number less than 5. There are four outcomes that fulfill this condition: rolling a 1, 2, 3, or 4. So the probability is \(\frac{4}{6} = \frac{2}{3}\).
04

Probability of a rolling a number no greater than 3

The event is rolling a number no greater than 3. There are three outcomes that fulfill this condition: rolling a 1, 2, or 3. So the probability is \(\frac{3}{6}= \frac{1}{2}\).

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Key Concepts

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

Probability of single events
When we talk about probability, especially in the context of single events, we're looking at the chance of one specific outcome happening. Imagine rolling a six-sided die. Each side represents a possible outcome. The probability of a particular outcome, such as rolling a 3, is the number of ways that outcome can occur divided by the total number of possible outcomes. Here, since only one of the six outcomes is a 3, the probability is \(\frac{1}{6}\).
This principle applies to any single event:
  • Identify the event whose probability you're finding.
  • Determine how many ways that event can happen.
  • Divide by the total number of possible outcomes.
This is one of the building blocks of understanding probability in more complex scenarios.
Basic probability concepts
Understanding probability means familiarizing oneself with a few fundamental concepts. At its core, probability quantifies the likelihood of an event happening, ranging from 0 to 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to occur.
Let's break down our exercise:
  • For an odd number, there are three favorable outcomes (1, 3, 5), out of six possibilities, giving a probability of \(\frac{3}{6} = \frac{1}{2}\).
  • Similarly, a number less than 5 has four favorable outcomes (1, 2, 3, 4), providing a probability of \(\frac{4}{6} = \frac{2}{3}\).
  • For a number no greater than 3, there are three outcomes (1, 2, 3), resulting in a probability of \(\frac{3}{6} = \frac{1}{2}\).
These examples underline the idea of favorable outcomes divided by possible outcomes, a fundamental concept in probability.
Mathematical problem-solving
Effective mathematical problem-solving in probability involves breaking down problems into clear, manageable steps. Here's how you can approach problems like the one we discussed:
1. **Identify the specific question:** What are you trying to find? For instance, whether you need the probability of rolling an odd number or a number less than 5. Each question requires a distinct approach.
2. **List all possible outcomes:** In the case of a die, the outcomes are 1 through 6. This determines your total potential set.
3. **Count favorable outcomes:** Determine which of those outcomes meet your condition. This is your numerator when calculating probability.
4. **Use the probability formula:** Place the number of favorable outcomes over the total number of outcomes. Simplify the fraction if possible.
This structured approach ensures that you systematically tackle each element of the problem, leading to a clear and correct solution.

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Most popular questions from this chapter

A company that manufactures shoes has three factories. Factory 1 produces \(25 \%\) of the company's shoes, Factory 2 produces \(60 \%,\) and Factory 3 produces \(15 \%\) One percent of the shoes produced by Factory 1 are mislabeled, \(0.5 \%\) of those produced by Factory 2 are mislabeled, and \(2 \%\) of those produced by Factory 3 are mislabeled. If you purchase one pair of shoes manufactured by this company, what is the probability that the shoes are mislabeled?

An experiment consists of two trials. The first is tossing a penny and observing whether it lands with heads or tails facing up; the second is rolling a die and observing a \(1,2,3,4,5,\) or 6 a. Construct the sample space using a tree diagram. b. List your outcomes as ordered pairs, with the first element representing the coin and the second, the die.

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A box contains marbles of five different colors: red, green, blue, yellow, and purple. There is an equal number of each color. Assign probabilities to each color in the sample space

A woman and a man (unrelated) each has two children. At least one of the woman's children is a boy, and the man's older child is a boy. Is the probability that the woman has two boys greater than, equal to, or less than the probability that the man has two boys? a. Demonstrate the truth of your answer by using a simple sample to represent each family. b. Demonstrate the truth of your answer by taking two samples, one from men with two-children families and one from women with two-children families. c. Demonstrate the truth of your answer using computer simulation. Using the Bernoulli probability function with \(p=0.5 \text { (let } 0=\text { girl and } 1=\text { boy })\), generate 500 "families of two children" for the man and the woman. Determine which of the 500 satisfy the condition for each and determine the observed proportion with two boys. d. Demonstrate the truth of your answer by repeating the computer simulation several times. Repeat the simulation in part c several times. e. Do the preceding procedures scem to yicld the same results? Explain.
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