Bayes' Theorem Solved Examples for University Students

📚 Quick Study Guide

• 🔢 Bayes' Theorem Formula: $P(A|B) = \frac{P(B|A) * P(A)}{P(B)}$
• 🔍 P(A|B): Posterior probability of event A occurring given that B has occurred.
• 🧪 P(B|A): Likelihood of event B occurring given that A has occurred.
• 📈 P(A): Prior probability of event A occurring.
• 📊 P(B): Prior probability of event B occurring.
• 💡 Total Probability: $P(B) = P(B|A)P(A) + P(B|A')P(A')$, where A' is the complement of A.
• 📝 Key Concept: Bayes' Theorem updates our beliefs (probability of an event) based on new evidence.

Practice Quiz

1. Question 1: A factory has two machines, X and Y, producing bolts. Machine X produces 60% of the bolts, and Machine Y produces 40%. 4% of the bolts produced by Machine X are defective, and 2% of the bolts produced by Machine Y are defective. If a bolt is selected at random and found to be defective, what is the probability that it was produced by Machine X?
   1. 0.667
   2. 0.75
   3. 0.545
   4. 0.6
2. Question 2: In a certain city, 30% of the people own a dog, and 10% own a cat. 5% of the people own both a dog and a cat. If a person is selected at random and found to own a dog, what is the probability that they also own a cat?
   1. 0.167
   2. 0.5
   3. 0.015
   4. 0.25
3. Question 3: A medical test has a sensitivity of 95% and a specificity of 90%. If the prevalence of a disease in a population is 1%, what is the probability that a person who tests positive actually has the disease?
   1. 0.087
   2. 0.95
   3. 0.1
   4. 0.5
4. Question 4: An email spam filter is 99% accurate in identifying spam. It also correctly identifies 95% of legitimate emails. If 80% of all emails are legitimate, what is the probability that an email flagged as spam is actually spam?
   1. 0.99
   2. 0.73
   3. 0.95
   4. 0.27
5. Question 5: Suppose you have two bags. Bag A contains 7 red balls and 3 blue balls. Bag B contains 2 red balls and 8 blue balls. You randomly pick a bag and then randomly pick a ball. If you pick a red ball, what is the probability it came from Bag A?
   1. 0.7
   2. 0.3
   3. 0.778
   4. 0.222
6. Question 6: A software company has two teams: Team Alpha and Team Beta. Team Alpha develops 70% of the software modules, and Team Beta develops the rest. 5% of the modules from Team Alpha have bugs, while 10% of the modules from Team Beta have bugs. If a module is chosen at random and found to have a bug, what is the probability it was developed by Team Alpha?
   1. 0.05
   2. 0.1
   3. 0.7
   4. 0.412
7. Question 7: A coin is flipped twice. Given that at least one of the flips resulted in heads, what is the probability that both flips resulted in heads?
   1. 1/2
   2. 1/3
   3. 1/4
   4. 2/3