Real World Examples of Conditional Probability for Algebra 2

📚 Quick Study Guide

• 🔍 Conditional Probability: The probability of an event A occurring, given that event B has already occurred. Denoted as $P(A|B)$.
• 🔢 Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(A \cap B)$ is the probability of both A and B occurring, and $P(B)$ is the probability of B occurring.
• 🎲 Independent Events: If A and B are independent, then $P(A|B) = P(A)$. The occurrence of B doesn't affect the probability of A.
• 📊 Real-World Examples: Think of medical tests, surveys, and quality control processes.
• 💡 Key Tip: Always identify the 'given' condition first. This is the event that has already happened.

🧪 Practice Quiz

1. A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability that the second ball is red, given that the first ball was blue?
   1. $\frac{5}{7}$
   2. $\frac{4}{7}$
   3. $\frac{5}{8}$
   4. $\frac{3}{7}$
2. In a class, 60% of the students like math, and 40% like science. 30% of the students like both math and science. What is the probability that a student likes science given that they like math?
   1. 0.5
   2. 0.4
   3. 0.3
   4. 0.2
3. A company finds that 70% of its customers are satisfied with their product. Of the satisfied customers, 80% would recommend the product to a friend. What percentage of all customers are both satisfied and would recommend the product?
   1. 56%
   2. 70%
   3. 80%
   4. 90%
4. A hospital tests patients for a disease. The test is 95% accurate. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease (approximately)?
   1. 19%
   2. 95%
   3. 1%
   4. 5%
5. Two dice are rolled. What is the probability that the sum is 7, given that at least one of the dice shows a 3?
   1. $\frac{1}{6}$
   2. $\frac{2}{11}$
   3. $\frac{1}{3}$
   4. $\frac{1}{4}$
6. A coin is flipped twice. What is the probability that both flips result in heads, given that the first flip is heads?
   1. $\frac{1}{2}$
   2. $\frac{1}{4}$
   3. 1
   4. 0
7. In a survey, 80% of people own a car, and 30% own a motorcycle. 20% own both. What is the probability that someone owns a motorcycle given they own a car?
   1. 0.25
   2. 0.20
   3. 0.30
   4. 0.35