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Ginger rolls a 6-sided die and then rolls it again. What is the probability that the first roll is at most 4 or the sum of the rolls is a multiple of 4?

[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline 1 & (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\
\hline 2 & (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\
\hline 3 & (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\
\hline 4 & (4,1) & (4,2) & (4,3) & (4,4) & (4,5) & (4,6) \\
\hline 5 & (5,1) & (5,2) & (5,3) & (5,4) & (5,5) & (5,6) \\
\hline 6 & (6,1) & (6,2) & (6,3) & (6,4) & (6,5) & (6,6) \\
\hline
\end{array}
\][/tex]


Sagot :

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].