Problem 4 - A First Course in Probability

Question

What is the probability that at least one of a pair of fair dice lands on \(6,\) given that the sum of the dice is \(i\) \(i=2,3, \ldots, 12 ?\)

Step-by-Step Solution

Answer

The probability of at least one die landing on 6, given the sum of the dice is \(i\) (where \(i = 2,3,\ldots,12\)), is as follows: | Sum i | Probability of at Least One 6 | |-------|-------------------------------| | 2 | 0 | | 3 | 0 | | 4 | 0.333 | | 5 | 0.500 | | 6 | 0.600 | | 7 | 0.667 | | 8 | 0.600 | | 9 | 0.500 | | 10 | 0.333 | | 11 | 0 | | 12 | 0 | The probability increases from 0 to 0.667 as \(i\) increases from 2 to 7 and decreases symmetrically as \(i\) increases from 8 to 12.

Step 1: Identify the Outcomes for a Given Sum i

For a given sum i, the possible outcomes can be represented as tuples (a,b), where a and b represent the outcomes of the first and second die, respectively. Since the dice are fair, both a and b can be any integer from 1 to 6. So, for each value of i (from 2 to 12), we need to identify all the tuples that result in the given sum i.

Step 2: Count the Total Number of Outcomes for a Given Sum i

To find the probability, we need to first count the total number of tuples for each value of i. It's important to note that for every sum i, there is a maximum value for a or b. We will use this to count the total outcomes for each i.

Step 3: Identify and Count the Favorable Outcomes for a Given Sum i

Now, we need to count the outcomes where at least one dice lands on 6. We will analyze each possible tuple for the given sum i, and check if either a or b is 6. We will count all such tuples for each sum i.

Step 4: Calculate the Probability for Each i

Finally, we will find the probability by dividing the number of favorable outcomes by the total number of outcomes for each value of i. We will round the probabilities to 3 decimal places for easier interpretation.

Step 5: Present the Results in a Table

To summarize the findings, we will present the calculated probabilities for each value of i in a table. The table will have 3 columns: "Sum i", "Total Outcomes", "Probability of at Least One 6". Here is the table with the calculated probabilities for each value of i: | Sum i | Total Outcomes | Probability of at Least One 6 | |-------|----------------|-------------------------------| | 2 | 1 | 0 | | 3 | 2 | 0 | | 4 | 3 | 0.333 | | 5 | 4 | 0.500 | | 6 | 5 | 0.600 | | 7 | 6 | 0.667 | | 8 | 5 | 0.600 | | 9 | 4 | 0.500 | | 10 | 3 | 0.333 | | 11 | 2 | 0 | | 12 | 1 | 0 | From the table, we can see that the probability of at least one die landing on 6 increases as the sum i increases from 2 to 7 and then decreases symmetrically as i further increases to 12.

Key Concepts

Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has already taken place. In the context of dice rolls, this concept allows us to calculate the probability of at least one die showing a 6, on the condition that the sum of the dice is a specific value, like 7 or 8. This is crucial when outcomes are dependent on previous events.

To illustrate, let's consider the sum of the dice is 7. The possible outcomes that total 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Among these, three outcomes meet our condition: one die showing a 6. Hence, if we already know that the sum is 7, the conditional probability of one die being a 6 is calculated by dividing the favorable outcomes (3) by the total outcomes (6).

Fair Dice Outcomes

When referring to 'fair' dice, we imply that each face of the die has an equal probability of landing face up. A standard six-sided die has outcomes ranging from 1 to 6, and each outcome has a 1 in 6 chance of occurring. This uniform likelihood is what makes the dice 'fair'. When rolling two dice, there are 36 possible outcomes (6 outcomes from the first die multiplied by 6 outcomes from the second die).

Understanding fair dice outcomes aids in setting the probabilities for different dice sums. For instance, the sum of 7 has six possible outcomes (see conditional probability section), while the sum of 2 or 12 only has one outcome each, which impacts the conditional probabilities that can be calculated.

Sum of Dice Probabilities

The sum of dice probabilities refers to the chance of rolling a particular sum with a pair of dice. Since dice sums can range from 2 to 12, understanding the distribution of these sums is key to analyzing dice roll probabilities.

For sums like 2 or 12, there is only one combination each: (1,1) and (6,6), respectively. As the sum increases from 2 towards 7, the number of combinations increases. At sum 7, which is the midpoint, there are the most combinations (six in total). Then, as the sum goes from 7 to 12, the number of combinations decreases symmetrically.

When calculating the probability of a sum, we take into account the total number of potential outcomes—always 36 for two six-sided dice—and the number of ways to achieve the specific sum. For instance, the probability of rolling a sum of 7 is calculated by the 6 favorable outcomes (that yield a sum of 7) divided by the total outcomes (36), resulting in a probability of 1/6.