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Probability Definition: A probability is the chance of some event, E, occurring in a specified manner. NOTATION: P{E} We can view probabilities from a Relative Frequency Interpretation P{E} =
# ways E occurs
# total events
Example 3.12 Consider the experiment where a fair coin is tossed twice. Find P{HH}. We call the collection of all possible events a sample space and we denote the sample space by . Definition Two events, E1 and E2 are disjoint (or mutually exclusive) if they cannot occur simultaneously Definition The union of two events, E1 and E2 is the event that E1 or E2 or both occur NOTATION: E1 ∪ E2 E1 or E2 Definition The intersection of two events, E1 and E2 is the event that E1 and E2 both occur NOTATION: E1 ∩ E2 E1 and E2 Definition Two events, E1 and E2 are independent if knowledge that E1 occurred does not affect P{E2} and vice versa. Two events that are not independent are dependent. Definition A conditional probability is the probability that one event occurs given that another event occurred. NOTATION: E1|E2 (reads “E 1 given E 2”) A common way to visualize how events interrelate in a sample space is through a Venn Diagram. Probability Rules Rule 1 0 ≤ P{E} ≤ 1 Rule 2 If E1, E2, … , Ek are all the possible (disjoint) events, ∑P{Ei} = 1 Rule 3 If Ec = {not E}, then P{Ec} = 1 – P{E} Rule 4 P{E1 ∪ E2} = P{E1} + P{E2} – P{E1 ∩ E2} Rule 5 If E1 and E2 are disjoint, then P{E1 ∪ E2} = P{E1} + P{E2} Rule 6 P{E1 ∩ E2} = P{E1|E2}P{E2} = P{E2|E1}P{E1} Rule 7 If E1 and E2 are independent, then P{E1 ∩ E2} = P{E1}P{E2} Probability Page 2 Examples Computing Probabilities Example 3.20 The following table summarizes the outcome of a study recording hair color and eye color for 1770 German men. For a randomly chosen male from this group, find P{Black Hair} P{Black or Red Hair} P{Black Hair or Blue Eyes} P{Red Hair and Brown Eyes} P{Blue Eyes | Black Hair} Probability Page 3 Tree Diagram A tree diagram is a useful tool for computing probabilities. Each branch in the tree diagram represents possible outcomes for a piece of an experiment. Probability Page 4 Example The ELISA (Enzyme‐Linked ImmunoSorbent Assay) is a common way to test for HIV infection. According to the CDC (the web page has since been unlinked) for the ELISA test, we have sensitivity = true positive = P{test positive | you have the disease} ≈ 99.9% specificity = true negative = P{testing negative | you don’t have the disease} ≈ 99.8% For a certain population, where 1/1000 are HIV positive, use a tree diagram to find: P{have HIV | test positive} and P{have HIV | test negative} Probability Page 5