Probability: Complements, Conditional Probability, and Bayes’ Theorem

Complements and the Probability of “At Least One”

Understanding Complements

In probability, the complement of an event is the event that the original event does not occur. The probability of “at least one” occurrence of an event is a common calculation in statistics, especially in quality control and reliability analysis.

• At least one means “one or more.”

• The complement of “at least one” is “none.”

Calculating the Probability of “At Least One”

1. Let $A$ = getting at least one of some event.

2. Let $\overline{A}$ = getting none of the event being considered.

3. Find $P(\overline{A})$ = probability that event $A$ does not occur.

4. Subtract the result from 1: $P(\text{at least one occurrence of event } A) = 1 - P(\text{no occurrences of event } A)$

Example: Manufacturing Defects

• A factory produces items with a 15% defect rate. If a customer buys 12 items, what is the probability of getting at least one defective item?

• Let $A$ = at least one defective item among 12.

• $\overline{A}$ = all 12 items are good (no defects).

• $P(\overline{A}) = 0.85^{12} = 0.142$

• $P(A) = 1 - 0.142 = 0.858$

Interpretation: There is an 85.8% chance of getting at least one defective item in a group of 12. This high probability suggests the need for improved quality control.

Conditional Probability

Definition and Notation

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

• Intuitive Approach: Assume event $A$ has occurred, then calculate the probability that $B$ will occur.

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

Example: Pre-Employment Drug Screening

Given the following table for 555 test subjects:

	Positive Test Result	Negative Test Result
Subject Uses Drugs	45 (True Positive)	5 (False Negative)
Subject Does Not Use Drugs	25 (False Positive)	480 (True Negative)

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

• b. $P(\text{subject uses drugs} | \text{positive test result}) = \frac{45}{70} = 0.643$

Interpretation: The probability that a drug user tests positive is 0.900, but the probability that a person who tests positive actually uses drugs is only 0.643. This demonstrates that $P(B|A) \neq P(A|B)$ in general.

Confusion of the Inverse

It is a common error to confuse $P(B|A)$ with $P(A|B)$. These probabilities are generally not equal, except in special cases.

Bayes’ Theorem

Introduction to Bayes’ Theorem

Bayes’ theorem is used to revise probability estimates based on new information. It is especially useful in medical testing and diagnostic applications.

Example: Interpreting Medical Test Results

• Prevalence of cancer: $P(C) = 0.01$

• False positive rate: $P(\text{positive test} | \text{no cancer}) = 0.10$

• True positive rate: $P(\text{positive test} | \text{cancer}) = 0.80$

Suppose 1000 subjects are tested:

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

• Total positive tests: $8 + 99 = 107$

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

Interpretation: Even with a positive test result, the probability of actually having cancer is only about 7.48%, much higher than the prior probability (1%), but still relatively low. This is due to the low prevalence and the false positive rate.

Prior and Posterior Probability

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

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

Bayes’ theorem formula:

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

This formula allows us to update our beliefs about the probability of an event based on new evidence.