Question
A group is playing a game that uses two number cubes labeled 1 to 6. A player rolls the number cubes and finds the sum of the numbers rolled. The set of all possible outcomes is shown in the table.
2
4
5
6
7
Possible Sums
3
4
5
6
7
8
4
5
6
7
8
9
5
6
6
7
7
8
8
9
9
10
10
11
7
8
9
10
11
12
If the sum is even and greater than 7, the player moves the number of spaces equal to the sum. Otherwise, the player does not move.
What is the probability that a player will move on a turn? Round the answer to the nearest hundredth. Enter the answer in the box.
2
4
5
6
7
Possible Sums
3
4
5
6
7
8
4
5
6
7
8
9
5
6
6
7
7
8
8
9
9
10
10
11
7
8
9
10
11
12
If the sum is even and greater than 7, the player moves the number of spaces equal to the sum. Otherwise, the player does not move.
What is the probability that a player will move on a turn? Round the answer to the nearest hundredth. Enter the answer in the box.
Answers
GPT-4o mini
To find the probability that a player will move based on the given conditions, we first need to identify the outcomes where the sum of the dice is even and greater than 7.
**Step 1: List all possible sums from rolling two number cubes:**
From the table, the possible sums when rolling two dice (number cubes) are:
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
**Step 2: Identify the sums that are even and greater than 7:**
The even sums greater than 7 from the list above are:
- 8
- 10
- 12
**Step 3: Count the favorable outcomes:**
Now let's count the number of ways to obtain these sums:
- The sum of **8** can occur in the following combinations:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
Total ways to get 8 = 5
- The sum of **10** can occur in the following combinations:
- (4, 6)
- (5, 5)
- (6, 4)
Total ways to get 10 = 3
- The sum of **12** can occur in the following combinations:
- (6, 6)
Total ways to get 12 = 1
Adding these together gives:
- Total favorable outcomes = 5 (for 8) + 3 (for 10) + 1 (for 12) = **9 favorable outcomes**
**Step 4: Calculate total possible outcomes:**
There are 6 faces on each die and hence:
- Total possible outcomes = 6 * 6 = **36**
**Step 5: Calculate the probability:**
The probability that a player will move is calculated as the ratio of favorable outcomes to total outcomes:
\[
P(\text{moving}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9}{36} = \frac{1}{4} = 0.25
\]
Rounded to the nearest hundredth, this is:
\[
\text{Probability} = \boxed{0.25}
\]
**Step 1: List all possible sums from rolling two number cubes:**
From the table, the possible sums when rolling two dice (number cubes) are:
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
**Step 2: Identify the sums that are even and greater than 7:**
The even sums greater than 7 from the list above are:
- 8
- 10
- 12
**Step 3: Count the favorable outcomes:**
Now let's count the number of ways to obtain these sums:
- The sum of **8** can occur in the following combinations:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
Total ways to get 8 = 5
- The sum of **10** can occur in the following combinations:
- (4, 6)
- (5, 5)
- (6, 4)
Total ways to get 10 = 3
- The sum of **12** can occur in the following combinations:
- (6, 6)
Total ways to get 12 = 1
Adding these together gives:
- Total favorable outcomes = 5 (for 8) + 3 (for 10) + 1 (for 12) = **9 favorable outcomes**
**Step 4: Calculate total possible outcomes:**
There are 6 faces on each die and hence:
- Total possible outcomes = 6 * 6 = **36**
**Step 5: Calculate the probability:**
The probability that a player will move is calculated as the ratio of favorable outcomes to total outcomes:
\[
P(\text{moving}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9}{36} = \frac{1}{4} = 0.25
\]
Rounded to the nearest hundredth, this is:
\[
\text{Probability} = \boxed{0.25}
\]