Statistics 3502/6304
Prof. Eric A. Suess
Chapter 4
Introduction to Probability
• A measure of uncertainty.
• Examples:
• Screening Tests – Pregnancy, Illegal Drugs, Sports Drug Tests
• Quality Control, Reliability, Six Sigma
• ESP
• Probability is the tool used to make inferences in Statistics
What does random mean?
• That was so random?
• This statement is past tense.
• An event is random before is happens. An outcome is random when
there are probabilities of the outcomes.
• Once the event has taken place it is no longer random. We can talk
about an outcome being likely or unlikely for a probability model.
Interpretations of Probability
• Classical Interpretation of Probability
• Based on counting
• Developed with games of chance.
• Flip a coin
• Draw a card from a shuffled deck of cards
• Blackjack, roulette, etc.
Interpretations of Probability
• Relative Frequency Interpretation of Probability
• Preform and experiment a large number of times and compute the
relative frequency of the event.
• Simulation
• Observing a production process
Interpretations of Probability
• Subjective Interpretation of Probability
• A one-time statement of the likelihood of an event occurring
• Weather prediction for tomorrow.
Finding Probabilities
• Using counting.
• Example: Flip a fair coin twice, what is the probability of HH? Of HT?
• Example: Draw a card from a shuffled deck of cards, what is the
probability of getting a Red card? Getting the King of Harts?
Events and Probability Rules
• The probability of an event A is between 0 and 1.
• The event A or B
• Two events are mutually exclusive if the occurrence of one event
excludes the possibility of the occurrence of the other event.
• Additive Rule for Mutually Exclusive Events: If two events, A and B,
are mutually exclusive, the probability of either event occurring is the
sum of the probability of each event.
Events and Probability Rules
• The complement of an event A is the event that A does not occur.
The complete of event A is 𝐴.
• The union of two events A and B is the set of all outcomes that are
included in either A or B (or both). 𝐴 𝐵
• The intersection of two events A and B is the set of all outcomes that
are included in both A and B. 𝐴 𝐵
• Additive Rule: Consider two event A and B, the probability of the
union of A and B is 𝑃(𝐴 𝐵) = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝐵)
Events and Probability Rules
• Conditional Probability: When a probability is computed knowing
the occurrence of another event.
• See Table 4.2
• Unconditional Probability: When a probability is computed overall.
• See Table 4.2
Events and Probability Rules
• The conditional probability of A given B is
𝑃(𝐴 𝐵)
𝑃 𝐴𝐵 =
𝑃(𝐵)
Events and Probability Rules
• The probability of the intersection of two events A and B is
𝑃(𝐴 𝐵) = 𝑃 𝐴 𝑃(𝐵|𝐴)
𝑃(𝐴 𝐵) = 𝑃 𝐵 𝑃(𝐴|𝐵)
• Independent events: 𝑃 𝐴 𝐵 = 𝑃 𝐴
• Dependent events: 𝑃 𝐴 𝐵 ≠ 𝑃 𝐴
• Multiplication Rule for Independent events: If A and B are
independent, the 𝑃(𝐴 𝐵) = 𝑃 𝐴 𝑃(𝐵)
Events and Probability Rules
• Bayes’ Rule
• False Positive
• False Negative
• Sensitivity
• Specificity
• Prevalence, prior probability
• Posterior probability
• Example 4.3 and 4.4