Ethan rolls the number cube and then spins the arrow on the spinner. A list of all the possible outcomes is shown.

Question

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

Answer 1

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