Math Problem Statement
Consider the following. (Assume that what is observed are the numbers that face up.) HINT [See Examples 1-3.] Two indistinguishable dice are rolled; the numbers add to 6. Describe the sample space S of the experiment. (Select all that apply.)
(2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (6,6) (3,3) (3,4) (3,5) (3,6) (5,5) (5,6) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (4,4) (4,5) (4,6) (4,4) (4,5) (4,6) (4,7) (7,7) (2,2) (2,3) (2,4) (2,5) (2,6) (6,6) (6,7) (3,3) (3,4) (3,5) (3,6) (3,7) (5,5) (5,6) (5,7)
List the elements of the given event. (Select all that apply.)
(4,5) (3,6) (1,2) (4,4) (3,5) (2,6) (5,5) (5,6) (6,6) (3,3) (2,4) (4,2) (1,5) (5,1) (3,4) (2,5) (1,6) (3,3) (2,4) (1,5)
Solution
The problem involves rolling two indistinguishable dice and finding the sample space when the sum of the two numbers is 6. Let's go step by step to describe the sample space and the elements of the event where the numbers add to 6.
Step 1: Possible combinations where the sum is 6
We are looking for all pairs of numbers on two dice that sum to 6. Since the dice are indistinguishable, the order of the numbers does not matter (i.e., (1,5) is the same as (5,1)).
Here are the combinations that sum to 6:
- (1,5)
- (2,4)
- (3,3)
- (4,2)
- (5,1)
Step 2: Filtering valid pairs
From the given list, we can now filter out the relevant pairs where the sum is 6:
Valid pairs:
- (1,5)
- (2,4)
- (3,3)
- (4,2)
- (5,1)
These are all the elements of the event where the numbers on the dice add to 6.
Step 3: Listing the elements of the event
From the options provided, the correct selections would be:
- (1,5)
- (2,4)
- (3,3)
- (4,2)
- (5,1)
These are the only valid pairs where the sum is 6.
Would you like further details or clarification? Here are five related questions you could explore:
- How would the sample space change if the dice were distinguishable?
- What is the probability of getting a sum of 6 when rolling two indistinguishable dice?
- What if the sum were 7 instead of 6? How would the sample space change?
- How would the calculation differ if you were rolling three dice?
- What is the total number of possible outcomes when rolling two indistinguishable dice?
Tip: When dealing with indistinguishable objects (like these dice), the order does not matter, so always be careful not to count duplicates.
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Sample Space
Formulas
Sum of two numbers on dice = 6
Theorems
Basic probability principles for calculating sample spaces
Suitable Grade Level
Grades 6-8