Chapter 4: Probability – Complements, Conditional Probability, and Bayes’ Theorem

Probability: Complements, Conditional Probability, and Bayes’ Theorem

Overview

This chapter introduces advanced concepts in probability, focusing on the use of complements to find probabilities of "at least one" events, the calculation and interpretation of conditional probabilities, and an introduction to Bayes’ Theorem for updating probabilities with new information.

Basic Concepts of Probability

Probability quantifies the likelihood of an event occurring, with values ranging from 0 (impossible) to 1 (certain). Fundamental rules include the addition and multiplication rules, which are extended in this chapter to more complex scenarios.

Addition and Multiplication Rules

The addition rule is used to find the probability that at least one of several events occurs. The multiplication rule is used to find the probability that multiple events all occur, especially in independent trials.

Complements: The Probability of “At Least One”

Definition and Key Ideas

• At least one means "one or more" occurrences of an event.

• The complement of getting "at least one" occurrence is getting no occurrences of the event.

General Formula

To find the probability of getting at least one occurrence of an event A in several trials:

• Let A = getting at least one occurrence of the event.

• The complement, \( \overline{A} \), is getting none of the event.

The probability is calculated as:

$P(\text{at least one occurrence of event } A) = 1 - P(\text{no occurrences of event } A)$

Example: Accidental iPad Damage

• A study found that 6% of damaged iPads were damaged by bags/backpacks.

• If 20 damaged iPads are randomly selected, what is the probability that at least one was damaged in a bag/backpack?

Solution Steps:

1. Let A = at least 1 of the 20 iPads was damaged in a bag/backpack.

2. The complement \( \overline{A} \) = all 20 iPads were damaged in a way other than bag/backpack.

3. Find \( P(\overline{A}) \): $P(\overline{A}) = 0.94^{20} = 0.290$

4. Find \( P(A) \): $P(A) = 1 - P(\overline{A}) = 1 - 0.290 = 0.710$

Interpretation: In a group of 20 damaged iPads, there is a 0.71 probability of getting at least one iPad damaged in a bag/backpack. This probability is not very high, so to be reasonably sure, more than 20 iPads should be used.

Conditional Probability

Definition

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as \( P(B|A) \), the probability of event B given event A.

Calculation Methods

• Intuitive Approach: Assume event A has occurred, then calculate the probability of B within that context.

• Formal Approach: Use the formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

Example: Pre-Employment Drug Screening

Suppose a table summarizes drug test results for 555 subjects:

	Positive Test Result	Negative Test Result
Subject Uses Drugs	45	5
Subject Does Not Use Drugs	25	480

• Find \( P(\text{positive test} | \text{subject uses drugs}) \): $P(\text{positive test} | \text{uses drugs}) = \frac{45}{50} = 0.900$

• Find \( P(\text{uses drugs} | \text{positive test}) \): $P(\text{uses drugs} | \text{positive test}) = \frac{45}{70} = 0.643$

Interpretation:

• \( P(\text{positive test} | \text{uses drugs}) = 0.900 \): A subject who uses drugs has a 90% chance of a positive test.

• \( P(\text{uses drugs} | \text{positive test}) = 0.643 \): A subject with a positive test has a 64.3% chance of actually using drugs.

Confusion of the Inverse

In general, \( P(B|A) \neq P(A|B) \). Confusing these is called the confusion of the inverse.

• Example: Let D = "It is dark outdoors", M = "It is midnight". $P(D|M) = 1$ (It is always dark at midnight.) $P(M|D) \approx 0$ (It is rarely exactly midnight when it is dark.)

Bayes’ Theorem

Definition and Use

Bayes’ Theorem is used to revise probability estimates based on new information. It relates prior probability (initial estimate) and posterior probability (updated estimate after new evidence).

Formula

$P(A|B) = \frac{P(A) \times P(B|A)}{P(A) \times P(B|A) + P(\overline{A}) \times P(B|\overline{A})}$

Example: Interpreting Medical Test Results

• Prevalence of cancer: 1% (\( P(C) = 0.01 \))

• True positive rate: 80% (\( P(\text{positive} | C) = 0.80 \))

• False positive rate: 10% (\( P(\text{positive} | \overline{C}) = 0.10 \))

Assume 1000 subjects:

	Positive Test Result	Negative Test Result	Total
Cancer	8	2	10
No Cancer	99	891	990

• \( P(C | \text{positive test}) = \frac{8}{107} = 0.0748 \)

Interpretation: Even with a positive test, the probability of actually having cancer is only about 7.5%, much higher than the prior probability (1%), but still not certain.

Prior and Posterior Probability

• Prior probability: The initial probability before new evidence (e.g., \( P(C) = 0.01 \)).

• Posterior probability: The updated probability after considering new evidence (e.g., \( P(C | \text{positive test}) = 0.0748 \)).

Summary Table: Key Terms and Formulas

Term	Definition
Complement	The event that an event does not occur.
Conditional Probability	Probability of one event given another has occurred.
Prior Probability	Initial probability before new evidence.
Posterior Probability	Updated probability after new evidence.
Bayes’ Theorem	Formula for updating probabilities with new information.

Key Formulas

• At least one: $P(\text{at least one}) = 1 - P(\text{none})$

• Conditional probability: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

• Bayes’ Theorem: $P(A|B) = \frac{P(A) \times P(B|A)}{P(A) \times P(B|A) + P(\overline{A}) \times P(B|\overline{A})}$