Question

Sam and Silvie go bowling. The following table shows how many total pins they knocked down in each of the 10 frames.

Sam: 5 l 6 l 7 l 8 l 10 l 9 l 4 l 10 l 4 l 6
Silvie: 8 l 5 l 7 l 5 l 2 l 9 l 10 l 4 l 5 l 9

If the sample space is all of the possible numbers of pins that can be knocked down in a single frame, which subset would be considered a complement of the set of pins that Sam and Silvie knocked down?

a. {2,6}
b. {2,4,5,6,7,9,10}
c. {1,2,3,4,5,6,7,8,9,10}
d. {0,1,3}

Answer 1

To find the complement of the set of pins that Sam and Silvie knocked down, we first need to identify the set of unique pin combinations based on their scores in the 10 frames.

Sam's scores are: 5, 6, 7, 8, 10, 9, 4, 10, 4, 6
Silvie's scores are: 8, 5, 7, 5, 2, 9, 10, 4, 5, 9

Now let's combine these scores and find the unique values:

- Sam: 5, 6, 7, 8, 10, 9, 4 (total unique pins from Sam's scores)
- Silvie: 8, 5, 7, 5, 2, 9, 10, 4 (total unique pins from Silvie's scores)

Taking the unique values from both players, we get:

{2, 4, 5, 6, 7, 8, 9, 10}

Now, the sample space is all possible numbers of pins that can be knocked down in a single frame, which is typically from 0 to 10:

Sample space: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The complement of the set of pins that Sam and Silvie knocked down (which is {2, 4, 5, 6, 7, 8, 9, 10}) consists of the numbers in the sample space that are **not** in their combined set.

Thus, the complement set is:

{0, 1, 3}

Now we can evaluate the options provided:

a. {2, 6} - Not the complement.
b. {2, 4, 5, 6, 7, 9, 10} - Not the complement; these are the numbers they knocked down.
c. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - Not the complement; it contains numbers they knocked down.
d. {0, 1, 3} - This is indeed the complement of the set of pins knocked down by Sam and Silvie.

Therefore, the correct answer is:

**d. {0, 1, 3}**