Problem 6
Question
A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is at least 4 .
Step-by-Step Solution
Verified Answer
The probability of getting a sum of at least 4 when rolling a pair of dice is \(\frac{11}{12}\).
1Step 1: Find the total number of possible outcomes.
The total number of possible outcomes when two dice are rolled can be found by multiplying the number of sides on one die (6) by the number of sides on the other die (6). The total number of possible outcomes is \(6 \times 6 = 36\).
2Step 2: Identify the successful outcomes (sum at least 4).
First, we will list all the possible outcomes with their sums:
(1, 1): 2
(1, 2): 3
(1, 3): 4
(1, 4): 5
(1, 5): 6
(1, 6): 7
(2, 1): 3
(2, 2): 4
(2, 3): 5
(2, 4): 6
(2, 5): 7
(2, 6): 8
(3, 1): 4
(3, 2): 5
(3, 3): 6
(3, 4): 7
(3, 5): 8
(3, 6): 9
(4, 1): 5
(4, 2): 6
(4, 3): 7
(4, 4): 8
(4, 5): 9
(4, 6): 10
(5, 1): 6
(5, 2): 7
(5, 3): 8
(5, 4): 9
(5, 5): 10
(5, 6): 11
(6, 1): 7
(6, 2): 8
(6, 3): 9
(6, 4): 10
(6, 5): 11
(6, 6): 12
We will now count the number of outcomes where the sum is at least 4:
(1, 3): 4
...
(6, 6): 12
This gives us a total of 33 successful outcomes.
3Step 3: Calculate the probability.
To calculate the probability, divide the number of successful outcomes (33) by the total number of possible outcomes (36).
\(P(sum \ge 4) = \frac{33}{36}\)
Now we simplify:
\(P(sum \ge 4) = \frac{11}{12}\)
So, the probability of getting a sum of at least 4 when rolling a pair of dice is \(\frac{11}{12}\).
Key Concepts
Discrete MathematicsCombinatoricsMathematical Reasoning
Discrete Mathematics
Discrete Mathematics is a branch of mathematics dealing with discrete elements that use distinct values. In this context, we focus on calculating the probability of outcomes involving dice. When rolling two dice, each die can result in a number from 1 to 6. Unlike continuous data that can take any value in a range, these outcomes are specific and separate.
Discrete mathematics often deals with structures like graphs, integers, and statements in logic that are countable or finite. In our problem, the number of possible outcomes is finite — 36 possible outcomes when rolling two dice. Each is a pair representing the numbers on both dice: (1,1), (1,2), up to (6,6).
Understanding these separate and countable outcomes is essential in discrete mathematics, as this underpins many of its applications in real-life scenarios and in computer science, where discrete elements often model networks or data structures.
Discrete mathematics often deals with structures like graphs, integers, and statements in logic that are countable or finite. In our problem, the number of possible outcomes is finite — 36 possible outcomes when rolling two dice. Each is a pair representing the numbers on both dice: (1,1), (1,2), up to (6,6).
Understanding these separate and countable outcomes is essential in discrete mathematics, as this underpins many of its applications in real-life scenarios and in computer science, where discrete elements often model networks or data structures.
Combinatorics
Combinatorics is the mathematical study of counting and arranging. It plays a big role in probability because it helps us systematically count possible outcomes in any given situation.
In our dice problem, combinatorics helps identify all unique pairs of numbers that result from rolling two dice. By determining that there are 36 possible outcomes (since each die has 6 faces, i.e., 6 * 6 = 36), we set a basis for calculating probabilities.
We use combinatorial methods to count the cases where the sum of numbers on the dice is at least 4. We list outcomes sequentially, starting from the first scenario where a sum of 4 is possible: (1,3) all the way up to (6,6). This approach ensures we account for all successful outcomes (33 in total) without missing any, illustrating how combinatorics provides a framework for organizing and counting complex sets of data skillfully.
In our dice problem, combinatorics helps identify all unique pairs of numbers that result from rolling two dice. By determining that there are 36 possible outcomes (since each die has 6 faces, i.e., 6 * 6 = 36), we set a basis for calculating probabilities.
We use combinatorial methods to count the cases where the sum of numbers on the dice is at least 4. We list outcomes sequentially, starting from the first scenario where a sum of 4 is possible: (1,3) all the way up to (6,6). This approach ensures we account for all successful outcomes (33 in total) without missing any, illustrating how combinatorics provides a framework for organizing and counting complex sets of data skillfully.
Mathematical Reasoning
Mathematical reasoning involves logical thinking to solve problems and prove statements. In the context of this exercise, we use mathematical reasoning to compute the probability of obtaining a sum of at least 4 when rolling a pair of dice.
We start by understanding the problem: finding out how many of the total possible outcomes (36) result in a sum of at least 4. Then, we reason through each of the combinations, identifying successful outcomes and confirming that they meet the criteria.
Lastly, applying the formula for probability, \[P = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}\],ensures our results are accurate and logical. Thus, slicing down the 33 successful outcomes out of 36 possible leads to a probability of \(\frac{11}{12}\). This logical flow from understanding, counting, to applying the formula underscores the importance of structured thinking in mathematical reasoning.
We start by understanding the problem: finding out how many of the total possible outcomes (36) result in a sum of at least 4. Then, we reason through each of the combinations, identifying successful outcomes and confirming that they meet the criteria.
Lastly, applying the formula for probability, \[P = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}\],ensures our results are accurate and logical. Thus, slicing down the 33 successful outcomes out of 36 possible leads to a probability of \(\frac{11}{12}\). This logical flow from understanding, counting, to applying the formula underscores the importance of structured thinking in mathematical reasoning.
Other exercises in this chapter
Problem 5
Verify directly that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\) for the sets in Exercise \(3 .\)
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List the elements of the set in roster notation. $$ \\{x \mid x \text { is a digit in the number } 352,646\\} $$
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Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experim
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