Probability Basics 1Fd065

1. **Stating the problem:** Probability is the branch of mathematics that studies the likelihood of events occurring. 2. **Formula and rules:** The probability $P$ of an event $E$ is given by the formula: $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$ Important rules: - $0 \leq P(E) \leq 1$ - The sum of probabilities of all possible outcomes is 1. - For mutually exclusive events $A$ and $B$, $P(A \cup B) = P(A) + P(B)$. - For independent events $A$ and $B$, $P(A \cap B) = P(A) \times P(B)$. 3. **Intermediate work and explanation:** Suppose you roll a fair six-sided die. The total number of possible outcomes is 6. - Probability of rolling a 4: $$P(4) = \frac{1}{6}$$ - Probability of rolling an even number (2, 4, or 6): $$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$ 4. **Learner-friendly explanation:** Probability measures how likely something is to happen. If an event is certain, its probability is 1. If it cannot happen, its probability is 0. For example, when flipping a fair coin, the chance of getting heads is $\frac{1}{2}$ because there are 2 possible outcomes and only 1 favorable outcome. 5. **Interactive aspect:** You can try calculating probabilities for different scenarios, such as drawing cards from a deck or picking colored balls from a bag, using the formula above. This introduction covers the basics of probability with formulas, rules, and examples to help you understand and apply the concept.