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Chapter 22: Problem 49

If you toss two dice, what is the total number of ways in which you can obtain (a) a 12 and (b) a \(7 ?\)

Short Answer

Expert verified
There is 1 way to roll a sum of 12 and 6 ways to roll a sum of 7 with two dice.

Step by step solution

01

Analysis of the Dice

There are two dice and each has 6 faces. Therefore, the number of possible outcomes when two dice are rolled together is \(6 * 6 = 36.\)
02

Find the Ways to Sum to 12

A 12 can only be achieved by rolling a 6 on each die. Thus, there is only 1 way to roll a 12 out of 36 possibilities.
03

Find the Ways to Sum to 7

There are several ways to roll a 7: (1,6); (2,5); (3,4); (4,3); (5,2); (6,1). Thus, there are 6 ways to roll a total sum of 7 out of 36 possibilities.

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics that deals with counting and arranging. It helps in finding out how many different combinations or ways exist to arrange a set of items. A key concept in combinatorics is understanding permutations and combinations:
  • Permutations: These are arrangements where order matters. For example, the permutations of the letters ABC are different from those of BAC.
  • Combinations: These are selections where order doesn't matter. For example, the combination of the letters ABC is the same as BAC.
When rolling two dice, combinatorics helps us calculate the total number of ways certain sums can be achieved. By knowing each die has 6 faces and the combinations they form, we determine different possible outcomes, such as exactly one instance of a sum of 12 or 6 instances for a sum of 7.
Dice Probabilities
Dice probabilities involve calculating the chance of certain outcomes when rolling dice. Each die has 6 faces, numbered from 1 to 6. The probability of any outcome is determined by dividing the number of successful outcomes by the total number of possible outcomes.
For two dice, each roll combination results in a total of 36 possible outcomes.
  • To calculate the probability of rolling a 12, realize that only one combination (6,6) works. Therefore, the probability is \(\frac{1}{36}\).
  • For a sum of 7, there are 6 favorable combinations: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Thus, the probability is \(\frac{6}{36} = \frac{1}{6}\).
Understanding dice probabilities helps in predicting and analyzing outcomes, useful in games and probability theory.
Sample Space
The sample space in probability refers to the set of all possible outcomes of an experiment. For two dice, each die being independent, a sample space would include 36 pairs, as each die's result can pair with every face of the other die.
Visualizing the sample space can be greatly simplified using a grid:
  • List the numbers from 1 to 6 for one die along both the rows and columns.
  • The intersections of these numbers form pairs that represent each possible outcome of the two dice.
This setup helps identify the pairs that yield desired sums like 12 or 7. Recognizing all distinct outcomes builds foundational understanding in organizing, visualizing, and dealing with probabilistic and combinatorial tasks efficiently.

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Most popular questions from this chapter

What is the maximum possible coefficient of performance of a heat pump that brings energy from outdoors at \(-3.00^{\circ} \mathrm{C}\) into a \(22.0^{\circ} \mathrm{C}\) house? Note that the work done to run the heat pump is also available to warm up the house.

A firebox is at \(750 \mathrm{K},\) and the ambient temperature is \(300 \mathrm{K}\) The efficiency of a Carnot engine doing \(150 \mathrm{J}\) of work as it transports energy between these constant-temperature baths is \(60.0 \% .\) The Carnot engine must take in energy \(150 \mathrm{J} / 0.600=250 \mathrm{J}\) from the hot reservoir and must put out \(100 \mathrm{J}\) of energy by heat into the environment. To follow Carnot's reasoning, suppose that some other heat engine \(S\) could have efficiency \(70.0 \% .\) (a) Find the energy input and wasted energy output of engine \(\mathrm{S}\) as it does \(150 \mathrm{J}\) of work. (b) Let engine S operate as in part (a) and run the Carnot engine in reverse. Find the total energy the firebox puts out as both engines operate together, and the total energy transferred to the environment. Show that the Clausius statement of the second law of thermodynamics is violated. (c) Find the energy input and work output of engine \(S\) as it puts out exhaust energy of \(100 \mathrm{J} .\) (d) Let engine S operate as in (c) and contribute \(150 \mathrm{J}\) of its work output to running the Carnot engine in reverse. Find the total energy the firebox puts out as both engines operate together, the total work output, and the total energy transferred to the environment. Show that the Kelvin-Planck statement of the second law is violated. Thus our assumption about the efficiency of engine \(\mathrm{S}\) must be false. (e) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. Show that the entropy statement of the second law is violated.

Heat engine \(X\) takes in four times more energy by heat from the hot reservoir than heat engine \(Y\). Engine \(X\) delivers two times more work, and it rejects seven times more energy by heat to the cold reservoir than heat engine \(Y\). Find the efficiency of (a) heat engine \(X\) and (b) heat engine \(Y\).

A sample consisting of \(n\) mol of an ideal gas undergoes a reversible isobaric expansion from volume \(V_{i}\) to volume \(3 V_{i} .\) Find the change in entropy of the gas by calculating \(\int_{i}^{t} d Q / T\) where \(d Q=n C_{P} d T\)

An electric power plant that would make use of the temperature gradient in the ocean has been proposed. The system is to operate between \(20.0^{\circ} \mathrm{C}\) (surface water temperature) and \(5.00^{\circ} \mathrm{C}\) (water temperature at a depth of about 1 \(\mathrm{km}\) ). (a) What is the maximum efficiency of such a system? (b) If the useful power output of the plant is \(75.0 \mathrm{MW},\) how much energy is taken in from the warm reservoir per hour? (c) In view of your answer to part (a), do you think such a system is worthwhile? Note that the "fuel" is free.
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