뒤로Probability: Complements, Conditional Probability, and Applications
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Probability
Complements: The Probability of “At Least One”
In probability theory, the concept of "at least one" is fundamental when determining the likelihood of an event occurring one or more times in a series of trials. Understanding complements is essential for efficiently solving these types of problems.
"At least one" means one or more occurrences of an event.
The complement of "at least one" is "none"—that is, the event does not occur at all.
Key Steps for Calculating the Probability of "At Least One" Occurrence:
Let A = getting at least one of some event.
The complement, denoted as \( \overline{A} \), is getting none of the event.
Find \( P(\overline{A}) \): the probability that event A does not occur.
Subtract this result from 1:
Example: Manufacturing Defects
A factory has a defect rate of 15%. If a customer purchases 12 products, what is the probability of getting at least one defective item?
Let A = at least one defective product among 12.
\( \overline{A} \) = all 12 products are good (no defects).
Probability all 12 are good: \( 0.85^{12} = 0.142 \)
Probability at least one is defective: \( 1 - 0.142 = 0.858 \)
Interpretation: There is an 85.8% chance of getting at least one defective product in a group of 12. This high probability suggests the manufacturing process needs improvement.
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 \( P(B \mid A) \), which reads as "the probability of B given A."
Intuitive Approach
To find \( P(B \mid A) \), assume event A has occurred and then calculate the probability that event B will occur under this condition.
Formal Approach
The formal formula for conditional probability is:
where \( P(A \text{ and } B) \) is the probability that both events A and B occur.
Example: Pre-Employment Drug Screening
The following table summarizes the results of a drug test study with 555 subjects:
Positive Test Result (Test shows drug use) | Negative Test Result (Test shows no drug use) | |
|---|---|---|
Subject Uses Drugs | 45 (True Positive) | 5 (False Negative) |
Subject Does Not Use Drugs | 25 (False Positive) | 480 (True Negative) |
Finding Conditional Probabilities
a. Probability of a positive test result given the subject uses drugs:
Assume the subject uses drugs (first row: 50 subjects).
45 had positive test results.
\( P(\text{positive test} \mid \text{uses drugs}) = \frac{45}{50} = 0.900 \)
Formal calculation:
\( P(\text{uses drugs and positive test}) = \frac{45}{555} \)
\( P(\text{uses drugs}) = \frac{50}{555} \)
\( P(\text{positive test} \mid \text{uses drugs}) = \frac{45/555}{50/555} = 0.900 \)
b. Probability the subject uses drugs given a positive test result:
Assume the subject had a positive test result (first column: 70 subjects).
45 of these use drugs.
\( P(\text{uses drugs} \mid \text{positive test}) = \frac{45}{70} = 0.643 \)
Interpretation
\( P(\text{positive test} \mid \text{uses drugs}) = 0.900 \): A drug user has a 90% chance of testing positive.
\( P(\text{uses drugs} \mid \text{positive test}) = 0.643 \): A person with a positive test has a 64.3% chance of actually using drugs.
In general, \( P(B \mid A) \neq P(A \mid B) \).
Additional info: The distinction between \( P(B \mid A) \) and \( P(A \mid B) \) is crucial in statistics and is often misunderstood. Conditional probability is foundational for advanced topics such as Bayes’ theorem and diagnostic testing.
