Conditional Probability and the Multiplication Rule

Conditional Probability and the Multiplication Rule

Conditional Probability

Conditional probability is a fundamental concept in probability theory, describing the likelihood of an event occurring given that another event has already occurred. Understanding conditional probability is essential for analyzing sequences of events and for distinguishing between independent and dependent events.

• Definition: The conditional probability of event B given event A is the probability that event B occurs under the condition that event A has already occurred. It is denoted as $P(B|A)$ and is read as "the probability of B, given A."

• Formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$ where $P(A \text{ and } B)$ is the probability that both events A and B occur, and $P(A)$ is the probability that event A occurs.

• Application: Conditional probability is used in real-world scenarios such as medical testing, quality control, and survey analysis.

Example: Two cards are drawn in sequence from a standard deck of 52 cards without replacement. The probability that the second card is a queen, given that the first card is a king, is:

• After removing a king, 51 cards remain, with 4 queens left.

• $P(\text{Queen on 2nd} | \text{King on 1st}) = \frac{4}{51} \approx 0.078$

Tabular Example: Survey Data

The following table summarizes survey results regarding adults who have ridden as a passenger in a self-driving vehicle:

Age	Yes	No	Total
18–64	202	497	699
65+	23	248	271
Total	225	745	970

Example: Probability that an adult is 18 to 64 years old, given that the adult has ridden as a passenger in a self-driving vehicle:

• There are 225 adults who answered "Yes"; 202 of them are 18–64 years old.

• $P(18\text{–}64 | \text{Yes}) = \frac{202}{225} \approx 0.898$

Try it Yourself: Find the probability that an adult is 65 or older, given that the adult has not ridden as a passenger in a self-driving vehicle.

Independent and Dependent Events

Understanding whether events are independent or dependent is crucial for applying probability rules correctly.

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

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

• Test for Independence: Events A and B are independent if $P(B|A) = P(B)$ or equivalently $P(A \text{ and } B) = P(A) \cdot P(B)$.

Example:

• Selecting a king (A) from a deck, not replacing it, then selecting a queen (B): Dependent

• Tossing a coin and getting a head (A), then rolling a die and getting a 6 (B): Independent

• Driving over 85 mph (A) and getting in a car accident (B): Dependent

Try it Yourself: Determine whether the following are independent or dependent: Smoking a pack of cigarettes per day and developing emphysema; Tossing a coin and getting a head, then tossing again and getting a tail.

Tabular Example: Monopoly Probabilities

The following table shows the probability of landing on certain Monopoly squares, depending on the length of a jail term:

Monopoly Square	Probability (Short Jail Term)	Probability (Long Jail Term)
Go	0.0310	0.0291
Chance	0.0087	0.0082
In Jail	0.0395	0.0946
Free Parking	0.0288	0.0283
Park Place	0.0219	0.0206
B&O RR	0.0307	0.0289
Water Works	0.0281	0.0265

Additional info: The probabilities change because the length of time spent in jail alters the number of turns a player is on the board, affecting the likelihood of landing on each square.

The Multiplication Rule

The Multiplication Rule is used to find the probability that two or more events occur in sequence. The rule differs depending on whether the events are independent or dependent.

• General Multiplication Rule: $P(A \text{ and } B) = P(A) \cdot P(B|A)$

• For Independent Events: $P(A \text{ and } B) = P(A) \cdot P(B)$

• Extension: For multiple independent events, multiply the probabilities of each event.

Example:

• Two cards drawn without replacement: Probability of king then queen: $P(\text{King then Queen}) = P(\text{King}) \cdot P(\text{Queen} | \text{King}) = \frac{4}{52} \cdot \frac{4}{51} \approx 0.006$

• Tossing a coin and rolling a die: Probability of head then 6: $P(\text{Head then 6}) = P(\text{Head}) \cdot P(6) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12} \approx 0.083$

Try it Yourself: Probability that two salmon swim successfully through a dam (each with probability 0.85): $0.85 \times 0.85 = 0.7225$

Using Complements with the Multiplication Rule

Sometimes, it is easier to find the probability of the complement of an event and subtract from 1. This is especially useful for "at least one" problems.

• Formula: Probability that at least one event occurs in n independent trials: $P(\text{at least one}) = 1 - P(\text{none})$

• For n independent events, each with probability of failure $q$: $P(\text{none succeed}) = q^n$

Example: Probability that at least one of three ACL surgeries is successful (each with $P(\text{success}) = 0.95$):

• Probability none are successful: $0.05^3 = 0.000125$

• Probability at least one is successful: $1 - 0.000125 = 0.999875$

Applications: Medical Residency Matching

Probability rules can be applied to real-world scenarios such as matching medical students to residency programs.

• In a given year, 19,326 U.S. MD seniors applied for residency; 18,108 were matched, and 75.6% of those matched got one of their top three choices.

• Probability a randomly selected senior was matched and got a top three choice: $P(\text{Matched and Top 3}) = P(\text{Matched}) \cdot P(\text{Top 3} | \text{Matched}) = \frac{18,108}{19,326} \cdot 0.756 \approx 0.708$

• Probability a matched senior did not get a top three choice: $1 - 0.756 = 0.244$

• Since 0.708 > 0.05, it is not unusual for a senior to be matched with a top three choice.

Summary Table: Key Probability Rules

aRule	Formula	When to Use
Conditional Probability	$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$	Probability of B given A has occurred
Multiplication Rule (General)	$P(A \text{ and } B) = P(A) \cdot P(B|A)$	Events may be dependent
Multiplication Rule (Independent)	$P(A \text{ and } B) = P(A) \cdot P(B)$	Events are independent
Complement Rule	$P(\text{at least one}) = 1 - P(\text{none})$	Finding probability of at least one event occurring

Additional info: In probability, events with probability less than 0.05 are typically considered unusual.