PSTAT 5A: Review Exercises
1. Basic Probability
A single fair six-sided die is rolled once.
1.1 What is \(P(\text{roll is an even number})\)?
1.2 What is \(P(\text{roll is 5 or 6})\)?
1. Basic Probability: Answers
1.1. \(P(\text{even}) = 3/6 = \tfrac{1}{2}\)
1.2. \(P(5 \text{ or } 6) = 2/6 = \tfrac{1}{3}\)
2. Joint & Conditional Probability
You draw two cards without replacement from a standard 52-card deck.
2.1. What is the probability that the first card is an Ace?
2.2. Given that the first card is an Ace, what is the probability that the second card is also an Ace?
2.3. Compute \(P(\text{both cards are Aces})\):
as a direct joint probability.
using conditional probability formula.
2. Joint & Conditional Probability: Answers
Solution. 2.1. \(4/52 = \tfrac{1}{13}\)
2.2. \(3/51 = \tfrac{1}{17}\)
2.3.
\(4/52 \times 3/51 = 1/221\)
\((1/13)\times(1/17) = 1/221\)
3. Independence vs. Mutual Exclusivity
Flip two fair coins in sequence. Define:
- \(A =\) first flip is Heads
- \(B =\) second flip is Heads
- \(C =\) both flips are Heads
3.1 Are events \(A\) and \(B\) independent?
3.2 Are events \(A\) and \(C\) mutually exclusive?
3.3 Compute \(P(A\cap B)\) and compare with \(P(A)P(B)\).
3. Independence vs. Mutual Exclusivity: Answers
Solution.
- Yes, independent: \(P(B|A)=1/2 = P(B)\)
- No, not mutually exclusive: \(A\cap C \neq \varnothing\).
- \(P(A\cap B)=1/4, \;P(A)P(B)=1/4\) → matches (independence).
4. Bonus Challenge
A bag contains 3 red balls and 2 blue balls. You draw one ball, replace it, then draw again.
4.1 Are the two draws independent? Why or why not?
4.2 Compute \(P(\text{red then blue})\).
4. Bonus Challenge: Answers
- Yes—they’re independent because of replacement.
- \((3/5)\times(2/5) = 6/25\)