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 4. Describe the sample space S of the experiment. (Select all that apply.)

(2,2) (2,3) (2,4) (2,5) (2,6) (4,4) (4,5) (4,6) (4,7) (4,4) (4,5) (4,6) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (5,5) (5,6) (5,7) (6,6) (6,7) (3,3) (3,4) (3,5) (3,6) (7,7) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (3,3) (3,4) (3,5) (3,6) (3,7) (6,6) (5,5) (5,6) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) Incorrect: Your answer is incorrect.

Solution

To solve this problem, let's analyze the situation where two indistinguishable dice are rolled, and their sum adds to 4. The key point is that the dice are indistinguishable, meaning the order of the numbers on the dice doesn't matter.

Step-by-step Analysis:

Possible dice outcomes that add to 4:

We are rolling two dice, and their sum must equal 4. These are the possible pairs of numbers that satisfy this condition:

  • (1, 3)
  • (2, 2)

The total number of outcomes is limited by the fact that dice are indistinguishable, so combinations like (3, 1) are equivalent to (1, 3), and we do not count it again.

Correct Sample Space SS:

The correct sample space for this experiment, where the numbers on the dice add to 4 and are indistinguishable, would only include:

  • (1, 3)
  • (2, 2)

This means our sample space consists of two pairs:
S={(1,3),(2,2)}S = \{ (1,3), (2,2) \}

Review of Incorrect Options:

Many of the options you provided contain combinations that either:

  1. Include sums greater than 4.
  2. List outcomes with numbers greater than 6, which is not possible for a six-sided die.
  3. Fail to account for the indistinguishability of the dice.

Thus, these extra outcomes are not part of the valid sample space for this problem.

Correct conclusion:

The correct description of the sample space is limited to the valid outcomes that add to 4, so you should focus only on the pairs (1,3) and (2,2).

Want more details or have any questions?

Here are 5 relative questions to expand this topic:

  1. What would the sample space be for the sum of 5 when rolling two indistinguishable dice?
  2. How does the sample space change when the dice are distinguishable?
  3. What are the odds of rolling a sum of 4 with two indistinguishable dice?
  4. How does symmetry affect the sample space in this type of dice experiment?
  5. What is the difference between calculating probabilities for distinguishable vs. indistinguishable dice?

Tip: When dealing with indistinguishable objects in probability, order does not matter, so make sure to count combinations rather than permutations.

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Math Problem Analysis

Mathematical Concepts

Probability
Combinatorics
Sample Space
Indistinguishable Objects

Formulas

Sum of outcomes formula for dice: number1 + number2 = target sum

Theorems

Combinatorial Counting Principle
Symmetry Principle in Probability

Suitable Grade Level

Grades 6-8

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