When rolling a pair of dice, there are two ways to get a point total of 3(1+2;2+1) but only one way to get a point total of 2(1+1). How many ways are there of getting point totals of 4 to 12? What is the most probable point total?

Verified step by step guidance

1. Identify all possible outcomes when rolling two dice. Each die has 6 faces, so calculate the total number of outcomes by multiplying the number of faces on one die by the number of faces on the other die.
2. List all possible sums from rolling two dice, which range from 2 to 12. For each sum, count the number of combinations of dice rolls that result in that sum. For example, a sum of 4 can be achieved by rolling (1,3), (2,2), or (3,1).
3. Calculate the number of ways to achieve each sum from 4 to 12 by considering the symmetry and possible combinations of dice rolls.
4. Compare the number of ways for each sum to determine which sum has the highest number of combinations. This sum will be the most probable point total because it has the most ways to occur.
5. Summarize the findings by listing the number of ways to achieve each sum from 4 to 12 and identifying the sum with the highest number of combinations as the most probable point total.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In the context of rolling dice, it helps determine how many combinations can result in a specific point total, allowing for the calculation of the most probable outcomes.

Combinatorial Counting

Combinatorial counting involves determining the number of ways to arrange or combine items, which is essential for calculating the different outcomes when rolling dice. For example, when rolling two dice, the total number of combinations is 6 (from the first die) multiplied by 6 (from the second die), resulting in 36 possible outcomes.

Sum of Dice Rolls

The sum of dice rolls refers to the total value obtained when adding the numbers shown on the faces of the rolled dice. For two six-sided dice, the possible sums range from 2 to 12, with varying frequencies for each total. Understanding how these sums are distributed helps identify the most probable outcomes based on the number of combinations that yield each total.