/*! 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 23 A pair of fair dice is rolled. W... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 23

A pair of fair dice is rolled. What is the probability that the second die lands on a higher value than does the first?

Short Answer

Expert verified
The probability that the second die lands on a higher value than the first one when rolling a pair of fair dice is \(\frac{5}{12}\), or approximately 0.4167 (rounded to four decimal places).

Step by step solution

01

Define the sample space S

The sample space S for rolling two dice can be found by considering all possible outcomes when rolling two dice. Since there are 6 faces on each die, there are a total of 6 x 6 = 36 possible outcomes when we roll them at the same time.
02

Define the event A - second die lands on a higher value than the first

Let's define event A as the second die shows a higher value than the first one. To find the probability of event A, we need to count the number of outcomes where the second die shows a higher value than the first die.
03

Count the number of favorable outcomes for event A

Here, we'll list the favorable outcomes where the second die shows a higher value than the first one: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 4), (2, 5), (2, 6) (3, 4), (3, 5), (3, 6) (4, 5), (4, 6) (5, 6) We have a total of 15 favorable outcomes for event A.
04

Calculate the probability of event A

Now that we have the total number of favorable outcomes for event A and the sample space size, we can calculate the probability of event A occurring. Recall that the probability of an event occurring is the ratio of the number of favorable outcomes to the total number of possible outcomes: \(P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{15}{36}\) To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3: \(P(A) = \frac{15}{36} = \frac{5}{12}\)
05

Interpret the result

The probability that the second die lands on a higher value than the first one when rolling a pair of fair dice is 5/12, or approximately 0.4167 (rounded to four decimal places). This means that there is approximately a 41.67% chance that the second die will show a higher value than the first one when rolling two fair dice.

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!

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

A forest contains 20 elk, of which 5 are captured, tagged, and then released. A certain time later, 4 of the 20 elk are captured. What is the probability that 2 of these 4 have been tagged? What assumptions are you making?

\(A, B,\) and \(C\) take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by$$ S=\left\\{\begin{array}{l}1,01,001,0001, \ldots, \\ 0000 \cdots\end{array}\right.$$ (a) Interpret the sample space. (b) Define the following events in terms of \(S:\) (i) \(A\) wins \(=A\) (ii) \(B\) wins \(=B\) (iii) \((A \cup B)^{c}\) Assume that \(A\) flips first, then \(B\), then \(C\), then \(A,\) and so on.

If two dice are rolled, what is the probability that the sum of the upturned faces equals \(i ?\) Find it for \(i=2,3, \ldots, 11,12\).

If a die is rolled 4 times, what is the probability that 6 comes up at least once?

Suppose that \(A\) and \(B\) are mutually exclusive events for which \(P(A)=.3\) and \(P(B)=.5 .\) What is the probability that (a) either \(A\) or \(B\) occurs? (b) \(A\) occurs but \(B\) does not? (c) both \(A\) and \(B\) occur?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

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.