2022-09-08
Independence
[PSDR] Definition 2.20
Two events are said to be independent if knowledge that one event occurs does not give any probabilistic information as to whether the other event occurs. Formally, we say that \(A\) and \(B\) are independent if \(P(A \cap B) = P(A)P(B)\).
Events \(A\) and \(B\) are said to be dependent if they are not independent.
Sampling WITH or WITHOUT replacement
Sampling WITH replacement: A method of sampling that an item is sampled more than once. Sampling with replacement generally produces independent events. Usually scenarios - but not always - like this does not distinguish identical items.
Sampling WITHOUT replacement: A method of sampling where an item may not be sampled more than once. Sampling without replacement generally produce dependent events. Usually - but not always - scenarios like this does distinguish identical items.
[PSDR] Definition 2.18
Let \(A\) and \(B\) be events in the sample space \(S\), with \(P(B) \ne 0\). The conditional probability of \(A\) given \(B\) is \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]
[PSDR] Proposition 2.1
Two simple facts about conditional probability are:
[PSDR] Theorem 2.3
Let \(A\) and \(B\) be events with non-zero probability in the sample space \(S\). The following are equivalent:
Consider a scenario where you draw three balls from an urn without replacement. The urn has 6 balls in total; 2 reds and 4 black.
What is the probability that at least two black balls are drawn?
Mini-Assignment: Independence and Conditional Probability
Back to Tentative Topics Schedule.