Question

The distribution of scores on dice. Suppose that you have \(n\) dice, each a different color, all unbiased and sixsided. (a) If you roll them all at once, how many distinguishable outcomes are there? (b) Given two distinguishable dice, what is the most probable sum of their face values on a given throw of the pair? (That is, which sum between 2 and 12 has the greatest number of different ways of occurring?) (c) What is the probability of the most probable sum?

Short answer

The number of distinguishable outcomes is \(6^n\). The most probable sum for two distinguishable dice is 7, with a probability of \( \frac{1}{6} \).

Step-by-step solution

Calculate the total number of distinguishable outcomes

Each die has 6 faces, and they are all different colors. Therefore, for each roll, there are 6 possible outcomes for each die. If there are n dice, multiply the number of possible outcomes for each die: Total outcomes = \(6^n\).

Determine the most probable sum for two dice

When calculating the possible sums of the face values between 2 and 12, note each individual sum's different combinations: Sum = 2: (1,1) Sum = 3: (1,2), (2,1) Sum = 4: (1,3), (2,2), (3,1) Continue this until all sums up to 12 have been considered. The sum with the most combinations is the most probable sum.

Find the most probable sum

Count the number of combinations for each sum: Sum = 2: 1 combination Sum = 3: 2 combinations Sum = 4: 3 combinations Sum = 5: 4 combinations Sum = 6: 5 combinations Sum = 7: 6 combinations Sum = 8: 5 combinations Sum = 9: 4 combinations Sum = 10: 3 combinations Sum = 11: 2 combinations Sum = 12: 1 combination The most probable sum is 7 because it has the most combinations – 6.

Calculate the probability of the most probable sum

Firstly, identify the total number of outcomes when rolling two dice: Total outcomes = \(6 \times 6 = 36\). Then use the number of combinations for the sum of 7: Probability = \( \frac{6}{36} = \frac{1}{6} \).

Key concepts

probability distribution

Probability distribution refers to the way in which probability is spread over different possible outcomes of a random experiment. In the context of rolling dice, we are interested in how likely each sum of the dice values is. When rolling two six-sided dice, each face has an equal chance (1/6) of landing face-up. This creates a distribution of probabilities across these sums. For example, the sum of 2 has only one combination (1-1), resulting in a lower probability. On the other hand, the sum of 7 has six combinations (1-6, 2-5, 3-4, etc.), making it the most probable sum. Understanding this distribution helps predict outcomes and determine the most likely results in similar experiments.

dice outcomes

When we talk about dice outcomes, we refer to all the possible results of rolling a set of dice. For a single six-sided die, there are 6 possible outcomes. Rolling multiple dice increases the number of possible outcomes exponentially. For n dice, there are \(6^n\) possible outcomes. For example, with two dice, the number of outcomes is \(6 \times 6 = 36\). These outcomes can be represented as ordered pairs, where the first element is the outcome of the first die and the second element is the outcome of the second die. If the dice are distinguishable by color, each combination remains unique. This is important for calculating probabilities and understanding the overall behavior of the rolls.

combinatorial analysis

Combinatorial analysis involves counting the number of ways certain events can occur. In the case of two dice, we're interested in how many ways each possible sum can be made. This involves listing all pairs of outcomes (from 1-1 to 6-6) and noting which sums they produce. For example, the sum of 7 can be achieved in 6 different ways: (1-6), (2-5), (3-4), (4-3), (5-2), and (6-1). Combinatorial analysis helps us determine that certain sums (like 7) are more probable because there are more combinations that result in that sum. This technique simplifies understanding the probability distribution across all potential outcomes.

most probable sum

The most probable sum is the sum that occurs the most frequently when rolling a set of dice. For two six-sided dice, the most probable sum is 7. This is because there are more combinations of dice rolls that result in 7 than any other number. Specifically, there are six combinations: (1-6), (2-5), (3-4), (4-3), (5-2), and (6-1). To find the probability of this sum, divide the number of favorable combinations by the total number of outcomes. In this case, \(\frac{6}{36} = \frac{1}{6}\). Understanding the most probable sum helps predict the outcomes of rolling dice, which is useful in games and probabilistic modelling.