4. Complement and union of events

Complementary events

Complementary event

Let $E$ be an event in a sample space $S$. The probability of event $\bar{E}$, or the complement of $E$ in $S$ is the sum of the probabilities of the outcomes not in $E$: $$ \boxed{ p(\bar E) = 1 - p(E) } $$

title: Example
We flip a fair coin $10$ times, what is the probability that heads comes up at least once?
 
Let $E$ be the event that heads comes up at least once of the $10$ flips.
Let $\bar E$ be the event that each of the $10$ times, tails comes up.
 
$$
	\begin{aligned}
		p(E) &= 1 - p(\bar E) \\
		&= 1 - \frac{1}{2^{10}} \\
		&= 1 - \frac{1}{1024} \\
		&= \frac{1023}{1024}
	\end{aligned}
$$

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Union of events

Union of events

The probability of the union of events is given by: $$ p(E_1 \cup E_2) = p(E_1) + p(E_2) - p(E_1 \cap E_2) $$

title: Example
What is the probability that a number randomly selected from $X = \{ n \in \mathbb N^+ | n \le 100 \}$ is divisible by either $2$ or $5$ or both (by $10$)?
 
- Let $E_1$ be the event that the selected number in $X$ is divisible by $2$.
- Let $E_2$ be the event that the selected number in $X$ is divisible by $5$.
- Then $E_1 \cup E_2$ is the event that it is divisible by both.
 
We are given that $|X| = 100, |E_1| = 50, |E_2| = 20, |E_1 \cap E_2| = 10$.
 
$$
	\begin{aligned}
		p(E_1 \cup E_2) &= \frac{50}{100} + \frac{20}{100} - \frac{10}{100} \\
		&= \frac{60}{100} \\
		&= \frac{3}{5} \\
		&= 0.6
	\end{aligned}
$$

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