1. **Stating the problem:** Define the following probability terms based on the context of rolling dice and flipping coins: Experiment, Sample Space, Event, Outcome, Probability, Mutually exclusive events, Independent events. 2. **Formulas and definitions:** - **Experiment:** A process or action that results in one or more outcomes. Here, rolling a die or flipping a coin. - **Sample Space ($S$):** The set of all possible outcomes. For a die, $S = \{1,2,3,4,5,6\}$; for a coin, $S = \{Heads, Tails\}$. - **Event ($E$):** A subset of the sample space, e.g., rolling an even number. - **Outcome:** A single result from the sample space, e.g., rolling a 4. - **Probability ($P$):** The measure of likelihood of an event, calculated as $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. - **Mutually exclusive events:** Events that cannot happen at the same time, e.g., rolling a 2 and rolling a 3 simultaneously. - **Independent events:** Events where the occurrence of one does not affect the probability of the other, e.g., flipping a coin and rolling a die. 3. **Explanation:** - An **experiment** is the action we perform to observe outcomes. - The **sample space** lists all possible outcomes. - An **event** is any collection of outcomes we are interested in. - Each **outcome** is a single possible result. - **Probability** quantifies how likely an event is to occur. - **Mutually exclusive events** cannot occur together. - **Independent events** have no influence on each other's outcomes. 4. **Example:** - Rolling a die and getting an even number is an event with outcomes $\{2,4,6\}$. - Flipping a coin and getting heads is an event with outcome $\{Heads\}$. - These two events are independent because the coin flip does not affect the die roll. Final answer: - Experiment: Rolling a die or flipping a coin. - Sample Space: $\{1,2,3,4,5,6\}$ for die; $\{Heads, Tails\}$ for coin. - Event: Any subset of outcomes, e.g., even numbers. - Outcome: A single result, e.g., 4. - Probability: $P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$. - Mutually exclusive: Events that cannot happen together. - Independent: Events where one does not affect the other.