Exam Questions on Applications of Counting in Probability (Statistics Course).

📚 Quick Study Guide

• 🔢 Fundamental Counting Principle: If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.
• 🎲 Permutations: Order matters! The number of permutations of $n$ objects taken $r$ at a time is given by: $P(n,r) = \frac{n!}{(n-r)!}$.
• 🧮 Combinations: Order doesn't matter! The number of combinations of $n$ objects taken $r$ at a time is given by: $C(n,r) = \frac{n!}{r!(n-r)!}$.
• 📌 Probability: The probability of an event A is given by: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$.
• 🤝 Independent Events: If events A and B are independent, then $P(A \text{ and } B) = P(A) \times P(B)$.
• ➕ Mutually Exclusive Events: If events A and B are mutually exclusive, then $P(A \text{ or } B) = P(A) + P(B)$.
• 💡 Conditional Probability: The probability of event A occurring given that event B has already occurred: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$.

Practice Quiz

1. Question 1: A restaurant offers 5 appetizers and 8 main courses. How many ways can a person order a two-course meal?
   1. 13
   2. 40
   3. 24
   4. 16
2. Question 2: How many different 4-letter arrangements can be formed from the letters of the word 'MATH', if each letter can only be used once?
   1. 12
   2. 24
   3. 120
   4. 360
3. Question 3: From a group of 6 men and 4 women, how many committees of 3 people can be formed consisting of 2 men and 1 woman?
   1. 60
   2. 80
   3. 120
   4. 180
4. Question 4: A bag contains 5 red balls and 3 blue balls. If two balls are drawn at random without replacement, what is the probability that both balls are red?
   1. 5/14
   2. 25/64
   3. 10/56
   4. 10/14
5. Question 5: A coin is flipped three times. What is the probability of getting at least two heads?
   1. 1/8
   2. 1/4
   3. 1/2
   4. 3/8
6. Question 6: Events A and B are independent. If P(A) = 0.3 and P(B) = 0.5, what is P(A and B)?
   1. 0.15
   2. 0.2
   3. 0.8
   4. 0.35
7. Question 7: In a class, 60% of the students like Math, and 40% like Science. If 20% like both, what percentage of students like either Math or Science?
   1. 100%
   2. 80%
   3. 60%
   4. 40%