Question

a fair die is a cube with number 1 through 6 on the sides, represented as painted spots. if a fair die is rolled, what is the probability of number 6 landing face up?

with a fair die, the probability of rolling any number 1 through 6 is the same.
if \\(p(1)\\) represents the probability of rolling a 1, \\(p(2)\\) the probability of rolling a 2, and so forth, what is the value of: \\(p(1) + p(2) + p(3) + p(4) + p(5) + p(6)\\)?

if two fair dice are rolled, there are two different ways to roll a sum of 3:
\

$$\begin{tabular}{|c|c|} \\hline die 1 & die 2 \\\\ \\hline 1 & 2 \\\\ \\hline 2 & 1 \\\\ \\hline \\end{tabular}$$

how many different ways are there to roll a sum of 4?

Explanation:

Identify the target sum and constraints

Using the Sample Space and Favorable Outcomes knowledge points
We need to find the number of different ways to roll a sum of \(4\) using two fair six-sided dice. Let the outcome of the first die be \(d_1\) and the second die be \(d_2\), where \(d_1, d_2 \in \{1, 2, 3, 4, 5, 6\}\). We seek pairs \((d_1, d_2)\) such that:

$$ d_1 + d_2 = 4 $$

List all favorable outcomes

Using the Favorable Outcomes knowledge point
We find all integer pairs \((d_1, d_2)\) within the range \([1, 6]\) that satisfy the equation:

• If \(d_1 = 1\), then \(d_2 = 3\). This gives the pair \((1, 3)\).
• If \(d_1 = 2\), then \(d_2 = 2\). This gives the pair \((2, 2)\).
• If \(d_1 = 3\), then \(d_2 = 1\). This gives the pair \((3, 1)\).
• If \(d_1 \ge 4\), \(d_2\) would have to be \(0\) or negative, which is impossible for standard dice.

Count the total number of ways

Using the Favorable Outcomes knowledge point
Counting the listed pairs:

1. \((1, 3)\)
2. \((2, 2)\)
3. \((3, 1)\)

There are exactly \(3\) different ways.

Answer:

How many different ways are there to roll a sum of 4? <blank>3</blank>