\begin{definition}[Dependent Events] Two events \$A\$ and \$B\$ are \textbf{dependent} if the occurrence of one event affects the probability of the other event occurring. \end{definition}
\begin{callout}[note] For \textbf{dependent events}, the probability of both events occurring is calculated using the formula:
\$\$ \Pr(A \text{ and } B) = \Pr(A) \cdot \Pr(B|A) \$\$
where \$\Pr(B|A)\$ is the \textbf{conditional probability} of \$B\$ occurring given that \$A\$ has occurred. \end{callout}
\begin{definition}[Conditional Probability] The \textbf{conditional probability} of an event \$B\$ given that another event \$A\$ has occurred is denoted by \$\Pr(B|A)\$ and is defined as:
\$\$ \Pr(B|A) = \frac{\Pr(A \text{ and } B)}{\Pr(A)} \$\$
provided that \$\Pr(A) \neq 0\$. \end{definition}
\begin{callout}[example] \textbf{Example 1: Calculating Probability of Dependent Events}
A box contains 5 red balls and 3 blue balls. Two balls are drawn one after the other without replacement. What is the probability that both balls are red?
\begin{enumerate} \item \textbf{Calculate the probability of drawing the first red ball:}
\\$\\$ \Pr(\text{First red}) = \frac{5}{8} \\$\\$ \item \textbf{Calculate the probability of drawing the second red ball given the first was red:} \\$\\$ \Pr(\text{Second red}|\text{First red}) = \frac{4}{7} \\$\\$ \item \textbf{Calculate the probability of both events occurring:} \\$\\$ \Pr(\text{First red and Second red}) = \Pr(\text{First red}) \cdot \Pr(\text{Second red}|\text{First red}) = \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14} \\$\\$
\end{enumerate}
The probability of drawing two red balls without replacement is \$\frac{5}{14}\$. \end{callout}
\begin{callout}[warning] \textbf{Common Mistake: Assuming Independence}
Students often mistakenly assume events are independent when they are not. Always check if the outcome of one event affects the other. \end{callout}
\begin{callout}[self_review] \begin{enumerate} \item A deck of cards has 52 cards. What is the probability of drawing two aces in a row without replacement?
\item A jar contains 10 red marbles and 5 green marbles. What is the probability of drawing a red marble followed by a green marble without replacement?
\end{enumerate} \end{callout}
\begin{callout}[tok] How does the concept of conditional probability reflect real-world situations where events are interconnected? Can you think of examples outside of mathematics where understanding dependencies is crucial? \end{callout}
