Question 1198683: A probability experiment consists of tossing a coin and then rolling a six-sided die. Determine the number of outcomes and identify the sample space.
II You roll a six-sided die. Find the probability of rolling numbers less than 5
III Differentiate between empirical probability and theoretical probability
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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.
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