Question

if two fair dice are rolled, there are two different ways to roll a sum of 3:
how many different ways are there to roll a sum of 4?
when rolling two fair dice, the probability of rolling different sums is
the same.
different.

Explanation:

Step1: Determine total number of outcomes

When rolling two dice, each die has 6 possible outcomes. So the total number of outcomes is \(6\times6 = 36\) (by the fundamental counting principle).

Step2: List pairs for sum of 4

Let the outcomes of the first die be \(x\) and the second die be \(y\). We want \(x + y=4\), where \(1\leq x\leq6\) and \(1\leq y\leq6\).
If \(x = 1\), then \(y=3\); if \(x = 2\), then \(y = 2\); if \(x=3\), then \(y = 1\). So the pairs \((1,3)\), \((2,2)\), \((3,1)\) give a sum of 4.

Step3: Analyze probability of different sums

The number of ways to get different sums varies. For example, sum of 2: \((1,1)\) (1 way), sum of 3: \((1,2),(2,1)\) (2 ways), sum of 4: 3 ways (as above). Since the number of ways to get each sum is not the same, the probability of rolling different sums (calculated as \(\frac{\text{number of ways for a sum}}{36}\)) is different.

Answer:

For the number of ways to roll a sum of 4: 3. For the probability question: B. different.