Question 1198683
**I. Number of Outcomes and Sample Space**

* **Number of Outcomes:** 
    * Coin Toss: 2 possible outcomes (Heads or Tails)
    * Die Roll: 6 possible outcomes (1, 2, 3, 4, 5, 6)
    * Total Outcomes: 2 (coin) * 6 (die) = 12 possible outcomes

* **Sample Space:** 
    * The sample space is the set of all possible outcomes. In this case:
        * {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} 
            * Where:
                * H = Heads 
                * T = Tails
                * 1-6 represents the numbers on the die

**II. Probability of Rolling Numbers Less Than 5**

* **Possible Outcomes:** 1, 2, 3, 4 
* **Number of Favorable Outcomes:** 4
* **Total Possible Outcomes:** 6
* **Probability:** 4 favorable outcomes / 6 total outcomes = 4/6 = 2/3 

**Therefore, the probability of rolling a number less than 5 on a six-sided die is 2/3.**

**III. Difference Between Empirical and Theoretical Probability**

* **Theoretical Probability:** 
    * Based on mathematical reasoning and assumptions about the fairness of the experiment (e.g., a fair coin and a fair die). 
    * It's the expected probability based on the ideal conditions.
    * Example: The theoretical probability of flipping heads on a fair coin is 1/2.

* **Empirical Probability:** 
    * Determined by conducting an experiment and observing the actual outcomes. 
    * It's based on the observed frequencies of events in a series of trials. 
    * Example: If you flip a coin 100 times and get 48 heads, the empirical probability of getting heads is 48/100 = 0.48.

**In Summary:**

* Theoretical probability is based on mathematical principles and assumptions.
* Empirical probability is based on actual observations from experiments.
* Empirical probability may not always perfectly match theoretical probability due to random chance and potential biases in the experiment.