Probability

Conditional Probability and the Multiplication Rule

This section explores the concept of conditional probability, the distinction between independent and dependent events, and the application of the multiplication rule to calculate probabilities of sequential events. These are foundational concepts in probability theory, essential for understanding more advanced statistical methods.

Conditional Probability

• Definition: Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as , which reads as "the probability of event B, given event A."

• Formula: The conditional probability of B given A is:

• Example: If two cards are drawn in sequence from a standard deck and the first card is a king (not replaced), the probability that the second card is a queen is , since there are 4 queens left in a deck of 51 cards.

• Application: Conditional probability is used in situations where the outcome of one event affects the likelihood of another.

Independent and Dependent Events

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, tossing a coin and rolling a die are independent events.

• Dependent Events: Events are dependent if the occurrence of one event changes the probability of the other. For example, drawing two cards from a deck without replacement.

• Examples:

  • Drawing a king and then a queen from a deck without replacement: Dependent

  • Tossing a coin and rolling a die: Independent

  • Driving over 85 mph and getting in a car accident: Dependent

The Multiplication Rule

• Definition: The multiplication rule is used to find the probability that two events, A and B, both occur in sequence.

• General Formula:

• For Independent Events:

• Extension: For any number of independent events:

• Example: The probability that three independent ACL surgeries are successful, each with probability 0.95:

• Complement Rule: The probability that at least one event occurs is:

Applications and Examples

• Medical Example: If the probability of success for ACL surgery is 0.95, the probability that none of three surgeries are successful is:

• Residency Matching Example: If 75.6% of matched medical students get one of their top three choices, the probability that a randomly selected senior is matched and gets a top three choice is:

• Complement Example: The probability that a matched senior does not get a top three choice is:

• Unusual Events: An event is considered unusual if its probability is less than 0.05. For example, a probability of 0.708 is not unusual.

Tabular Example: Conditional Probability from Survey Data

The following table summarizes survey results regarding adults who have ridden as passengers in self-driving vehicles:

To find the probability that an adult is 18 to 64 years old given they have ridden as a passenger in a self-driving vehicle:

Summary Table: Independent vs. Dependent Events