Problem 26 - Algebra for College Students

Question

In how many ways can a sum less than ten be obtained when tossing a pair of dice? 30

Step-by-Step Solution

Answer

There are 30 ways to obtain a sum less than 10 with two dice.

Step 1: Understanding the Problem

We need to determine the number of ways to get a sum of less than 10 when a pair of dice is tossed. Each die has numbers 1 to 6, so we need to consider all possible combinations of the dice outcomes that give sums from 2 to 9.

Step 2: List Possible Sums

The possible sums when rolling two dice can range from 2 (1+1) to 12 (6+6). We are interested in sums less than 10, so we need sums: 2, 3, 4, 5, 6, 7, 8, and 9.

Step 3: Calculate Combinations for Each Sum

Calculate how many combinations of die rolls result in each sum: - Sum 2: (1, 1) = 1 combination - Sum 3: (1, 2), (2, 1) = 2 combinations - Sum 4: (1, 3), (2, 2), (3, 1) = 3 combinations - Sum 5: (1, 4), (2, 3), (3, 2), (4, 1) = 4 combinations - Sum 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = 5 combinations - Sum 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 combinations - Sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5 combinations - Sum 9: (3, 6), (4, 5), (5, 4), (6, 3) = 4 combinations.

Step 4: Sum Up Combinations

Add up the number of combinations for all sums less than 10: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 = 30.

Step 5: Conclusion: Determine the Total Number of Ways

There are 30 different combinations of die rolls that result in a sum less than 10 when rolling a pair of dice.

Key Concepts

Combinatorial Analysis

Combinatorial analysis is a powerful mathematical tool used to count the number of ways certain events can occur. In the context of dice games, it helps us determine the different combinations that result from rolling two dice.
When we look at two dice, each die has six faces with numbers ranging from 1 to 6. The possible outcomes of throwing two dice can be represented as ordered pairs, resulting in a total of 36 possible combinations (since 6 possibilities for the first die times 6 possibilities for the second die). By using combinatorial analysis, we precisely compute how often each sum appears and evaluate the desired outcomes.
This approach doesn't just rely on listing possibilities randomly but instead uses a systematic counting method to ensure that no possibilities are overlooked. This is crucial in games of chance or probability-based problems to find accurate results.

Sum of Dice

When discussing the sum of dice, we refer to the total value obtained by adding the numbers on the top faces of the dice after their roll. For two six-sided dice, the smallest possible sum is 2 (both dice showing 1), and the largest is 12 (both dice showing 6).
The problem often involves finding the probability of obtaining a specific sum. To illustrate, when aiming for a sum of 7, there are several combinations: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), totaling six combinations. This frequency is important because some sums like 7 occur more frequently than others, such as 2 or 12, which only have one possible combination each.
Understanding the distribution of these sums allows one to predict expected outcomes and devise strategies based on likelihood in dice games. It illustrates the uneven probability distribution of sums that can result from two rolls.

Dice Outcomes

Dice outcomes refer to the possible results from throwing dice, and each combination represents a distinct outcome. With two dice, each rolled die can result in one of six outcomes, leading to 36 total outcomes (6 outcomes for the first die multiplied by 6 for the second). This array of outcomes captures all the possible scenarios when calculating probabilities in dice-based exercises.
For example, if interested in determining how often a sum is less than 10, we compute each possible sum from 2 to 9 and establish how many outcomes correspond to each. For sums of 7, which are frequent, there are six combinations, but for sums like 2, there's only one.
By considering each possible outcome, we methodically account for each scenario, ensuring our computed probabilities are comprehensive and accurate. Recognizing every distinct outcome helps in precisely identifying the statistical likelihood of each result in a dice game.