2023-10-02
Define independence of 2-3 events given probability notation
Calculate whether two or more events are independent
Question: Which of the following sequences of coin tosses of heads (\(H\)) and tails (\(T\)) is more likely to happen, assuming the coin is fair?
\[HTTHHHTHTHHTTTH\] or \[HTTTTTTTTHTTTTT\]
Definition: Independence
Events \(A\) and \(B\) are independent if \[\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B).\]
Notation: For shorthand, we sometimes write \[A \mathrel{\unicode{x2AEB}} B,\] to denote that \(A\) and \(B\) are independent events.
Example 1
Two dice (red and blue) are rolled. Let \(A =\) event a total of 7 appears, and \(B =\) event red die is a six. Are events \(A\) and \(B\) independent?
Definition: Independence of 3 Events
Events \(A\), \(B\), and \(C\) are mutually independent if
\(\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)\)
\(\mathbb{P}(A \cap C) = \mathbb{P}(A) \cdot \mathbb{P}(C)\)
\(\mathbb{P}(B \cap C) = \mathbb{P}(B) \cdot \mathbb{P}(C)\)
Remark:
On your homework you will show that \((1) \not \Rightarrow (2)\) and \((2) \not \Rightarrow (1)\).
Example 2
Suppose you take a random sample of \(n\) people, of which people are smokers and non-smokers independently of each other. Let
\(A_i =\) event person \(i\) is a smoker, for \(i=1, \ldots ,n\), and
\(p_i =\) probability person \(i\) is a smoker, for \(i=1, \ldots ,n\).
Find the probability that at least one person in the random sample is a smoker.