Here are the sample spaces for each of the scenarios you've provided:

1. The number of hours (to the nearest hour) of television watched yesterday by a student is recorded.

• Since the number is recorded to the nearest hour, we assume the possible values are integers from 0 hours upwards.
• Sample space: $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, \ldots\}$ (The upper limit would depend on the maximum number of hours a student could watch TV in one day, so this could be truncated based on context, e.g., $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ if considering a reasonable maximum of 8 hours.)

2. The management of an oil company must decide whether to go ahead with a new oil shale plant or to cancel it.

• This is a decision-making scenario with two possible outcomes: to go ahead or to cancel.
• Sample space: $S = \{ \text{Go Ahead}, \text{Cancel} \}$

3. A record is kept each day for three days about whether a particular stock goes up or down.

• The stock can either go up or down each day. Over three days, each day can have two outcomes.
• Sample space: $S = \{ \text{Up, Up, Up}, \text{Up, Up, Down}, \text{Up, Down, Up}, \text{Up, Down, Down}, \text{Down, Up, Up}, \text{Down, Up, Down}, \text{Down, Down, Up}, \text{Down, Down, Down} \}$ (This is all combinations of "Up" and "Down" over three days.)

4. A coin is tossed, and a die is rolled.

• The coin has two outcomes: heads or tails. The die has six outcomes: 1 through 6.
• Sample space: $S = \{ (\text{Heads}, 1), (\text{Heads}, 2), (\text{Heads}, 3), (\text{Heads}, 4), (\text{Heads}, 5), (\text{Heads}, 6), (\text{Tails}, 1), (\text{Tails}, 2), (\text{Tails}, 3), (\text{Tails}, 4), (\text{Tails}, 5), (\text{Tails}, 6) \}$

5. A box contains five balls, numbered 1, 2, 3, 4, and 5. A ball is drawn at random, the number on it recorded, and the ball replaced. The box is shaken, a second ball is drawn, and its number is recorded.

• Since the ball is replaced after each draw, the outcome of each draw is independent, and each draw has 5 possible outcomes (1 to 5).
• Sample space: $S = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5) \}$ (Each pair represents the outcome of the first and second draws.)

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Do you need further details or explanations on any of these scenarios? Let me know if you have any specific questions about the sample spaces!

Related questions:

1. How can the concept of sample space be used in probability calculations?
2. What is the difference between discrete and continuous sample spaces?
3. How does the concept of replacement affect sample spaces in probability?
4. How can you use sample spaces to calculate probabilities in experiments like tossing coins or rolling dice?
5. What are some real-life applications of sample spaces and probability theory?

Tip: Always consider the possible outcomes and how they relate to the scenario before constructing a sample space. The sample space should include every possible outcome of the experiment, no matter how unlikely.