Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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].
### 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].
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.