1. **Stating the problem:** We want to understand the Bernoulli distribution, which models a random experiment with exactly two possible outcomes: success or failure. 2. **Definition:** A Bernoulli distribution is a discrete probability distribution for a random variable $X$ which takes the value 1 (success) with probability $p$ and the value 0 (failure) with probability $1-p$. 3. **Probability mass function (PMF):** The formula for the Bernoulli distribution is: $$P(X=x) = p^x (1-p)^{1-x} \quad \text{for } x \in \{0,1\}$$ This means: - If $x=1$, then $P(X=1) = p$. - If $x=0$, then $P(X=0) = 1-p$. 4. **Important rules:** - $0 \leq p \leq 1$ because probabilities must be between 0 and 1. - The sum of probabilities for all possible outcomes is 1: $p + (1-p) = 1$. 5. **Expected value and variance:** - The expected value (mean) of $X$ is $E(X) = p$. - The variance of $X$ is $Var(X) = p(1-p)$. 6. **Interpretation:** The Bernoulli distribution is useful for modeling yes/no questions, coin flips, or any binary outcome. 7. **Example:** If a coin is fair, then $p=0.5$. The probability of heads (success) is 0.5, and tails (failure) is also 0.5. This completes the explanation of the Bernoulli distribution.