To determine the probability that the first roll is at most 4 or the sum of the rolls is a multiple of 4, we need to follow a structured approach.
### Step 1: Total Possible Outcomes
When rolling a 6-sided die twice, the number of total possible outcomes is [tex]\(6 \times 6 = 36\)[/tex].
### Step 2: Favorable Outcomes for the First Condition
Condition 1: The first roll is at most 4.
We can view this as:
- (1, x), (2, x), (3, x), (4, x)
- where [tex]\(x\)[/tex] can be any number from 1 to 6.
So, the number of favorable outcomes for this condition is:
[tex]\[ 4 \times 6 = 24 \][/tex]
### Step 3: Favorable Outcomes for the Second Condition
Condition 2: The sum of the rolls is a multiple of 4.
We now identify these pairs:
- (1, 3), (1, 7) (Only valid rolls go up to 6 so ignore (1, 7))
- (2, 2), (2, 6)
- (3, 1), (3, 5)
- (4, 4)
- (5, 3)
- (6, 2), (6, 6)
These pairs are:
- (1,3), (2,2), (2,6), (3,1), (3,5), (4,4), (5,3), (6,2), (6,6).
There are a total of 9 favorable outcomes for this condition.
### Step 4: Overlapping Outcomes for Both Conditions
We need to count the outcomes where both conditions are satisfied simultaneously (overlap).
For the pairs from the first condition (first roll at most 4):
- (1, x), (2, x), (3, x), (4, x)
We see which pairs with these first rolls also make the sum a multiple of 4:
- (1, 3)
- (2, 2), (2, 6)
- (3, 1), (3, 5)
- (4, 4)
These pairs are:
- (1, 3), (2, 2), (2, 6), (3, 1), (3, 5), (4, 4).
There are a total of 6 overlapping outcomes.
### Step 5: Calculate Total Favorable Outcomes
We combine both conditions subtracting the overlap to avoid double-counting:
[tex]\[ 24 + 9 - 6 = 27 \][/tex]
### Step 6: Calculate the Probability
The probability is the number of favorable outcomes divided by the total possible outcomes:
[tex]\[ \frac{27}{36} = \frac{3}{4} = 0.75 \][/tex]
Thus, the probability that the first roll is at most 4 or the sum of the rolls is a multiple of 4 is [tex]\(0.75\)[/tex] or [tex]\(75\%\)[/tex].
