Question
Ethan rolls the number cube and then spins the arrow on the spinner. A list of all the possible outcomes is shown.
StartSet the outcome 1 comma 1 the outcome 1 comma 2 the outcome 1 comma 3 the outcome 2 comma 1 the outcome 2 comma 2 the outcome 2 comma 3 the outcome 3 comma 1 the outcome 3 comma 2 the outcome 3 comma 3 the outcome 4 comma 1 the outcome 4 comma 2 the outcome 4 comma 3 the outcome 5 comma 1 the outcome 5 comma 2 the outcome 5 comma 3 the outcome 6 comma 1 the outcome 6 comma 2 the outcome 6 comma 3 EndSet
A number cube with 6 sides and a spinner divided into 3 equal parts.
Question
What is the probability that the sum of the two numbers is odd, given that the product is 6?
Answer options with 4 options
A.
1-third
B.
1-half
C.
2-thirds
D.
1
StartSet the outcome 1 comma 1 the outcome 1 comma 2 the outcome 1 comma 3 the outcome 2 comma 1 the outcome 2 comma 2 the outcome 2 comma 3 the outcome 3 comma 1 the outcome 3 comma 2 the outcome 3 comma 3 the outcome 4 comma 1 the outcome 4 comma 2 the outcome 4 comma 3 the outcome 5 comma 1 the outcome 5 comma 2 the outcome 5 comma 3 the outcome 6 comma 1 the outcome 6 comma 2 the outcome 6 comma 3 EndSet
A number cube with 6 sides and a spinner divided into 3 equal parts.
Question
What is the probability that the sum of the two numbers is odd, given that the product is 6?
Answer options with 4 options
A.
1-third
B.
1-half
C.
2-thirds
D.
1
Answers
GPT-4o mini
Let's analyze the possible outcomes based on the conditions provided: the product of the numbers rolled (from the number cube) and spun (from the spinner) must equal 6, and we want to find the probability that their sum is odd.
### Step 1: Finding Outcomes with Product 6
From the number cube (with values 1-6) and the spinner (with values 1-3), we can identify the combinations that yield a product of 6:
The pairs of outcomes (number cube, spinner) that provide a product of 6 are:
- (1, 6) → Not valid as the spinner only has 1-3.
- (2, 3) → Valid: \(2 \times 3 = 6\)
- (3, 2) → Valid: \(3 \times 2 = 6\)
- (6, 1) → Not valid as the number cube only has 1-6.
Hence, the valid outcomes for the product being 6 are:
1. (2, 3)
2. (3, 2)
### Step 2: Determine the Sums
Next, we calculate the sums of these valid outcomes:
1. \(2 + 3 = 5\) (odd)
2. \(3 + 2 = 5\) (odd)
Both valid outcomes yield an odd sum.
### Step 3: Computing the Probability
Now we calculate the probability that the sum is odd given the product is 6.
- Total outcomes with product 6: 2
- Outcomes with an odd sum: 2
Therefore, the probability that the sum is odd given that the product is 6 is:
\[
P(\text{sum is odd | product is 6}) = \frac{\text{Number of odd sum outcomes}}{\text{Total outcomes with product 6}} = \frac{2}{2} = 1
\]
### Conclusion
The final answer is:
**D. 1**
### Step 1: Finding Outcomes with Product 6
From the number cube (with values 1-6) and the spinner (with values 1-3), we can identify the combinations that yield a product of 6:
The pairs of outcomes (number cube, spinner) that provide a product of 6 are:
- (1, 6) → Not valid as the spinner only has 1-3.
- (2, 3) → Valid: \(2 \times 3 = 6\)
- (3, 2) → Valid: \(3 \times 2 = 6\)
- (6, 1) → Not valid as the number cube only has 1-6.
Hence, the valid outcomes for the product being 6 are:
1. (2, 3)
2. (3, 2)
### Step 2: Determine the Sums
Next, we calculate the sums of these valid outcomes:
1. \(2 + 3 = 5\) (odd)
2. \(3 + 2 = 5\) (odd)
Both valid outcomes yield an odd sum.
### Step 3: Computing the Probability
Now we calculate the probability that the sum is odd given the product is 6.
- Total outcomes with product 6: 2
- Outcomes with an odd sum: 2
Therefore, the probability that the sum is odd given that the product is 6 is:
\[
P(\text{sum is odd | product is 6}) = \frac{\text{Number of odd sum outcomes}}{\text{Total outcomes with product 6}} = \frac{2}{2} = 1
\]
### Conclusion
The final answer is:
**D. 1**