Lesson 1

Basic Problems.

1. (5 points) Three students work independently on the same problem; their success probabilities are \(0.8\), \(0.7\), and \(0.6\). What is the probability that at least one of them solves it?
   Hint:

   Use the complement: “none solve it”, then subtract from \(1\).
2. (5 points) It is known that events \(A\) and \(B\) are independent and mutually exclusive. Find \(\min(\Pr[A], \Pr[B])\).
3. (10 points) A student had \(10\) notebooks in a backpack: \(7\) squared and \(3\) lined. One notebook was forgotten at school. After that, one notebook is drawn uniformly at random from the backpack. What is the probability it is squared?
   Hint:

   Condition on which type was forgotten; then average.
4. (10 points) A drug test has \(99\%\) sensitivity (true positives for users) and \(98.5\%\) specificity (true negatives for non‑users). The prevalence of drug users is \(0.5\%\). Given a random test is positive, what is the probability the person is truly a user?
   Hint:

   Apply Bayes’ formula.
5. (10 points) Out of ten shooters: five hit a target with probability \(0.8\), three with probability \(0.5\), and two with probability \(0.9\). A randomly selected shooter fires and hits. To which group does the shooter most likely belong, and with what probability?
6. (10 points) Three coins: two are fair, the third has \(\Pr[\text{Heads}]=\frac{1}{3}\). One coin is chosen uniformly at random and flipped five times, yielding exactly one Head. Which kind of coin (fair vs. biased) is more likely, and by how much?
   Hint:

   Compare \(P_{\text{fair}}=\binom{5}{1}(1/2)^{5}\) vs. \(P_{\text{bias}}=\binom{5}{1}(1/3)(2/3)^{4}\) under priors \(2/3\) and \(1/3\).
7. (10 points) Consider a uniformly random permutation \(p=p_{1}p_{2}p_{3}\) of \([3]\). Let \(A\) be the event \(p_{1}>p_{2}\) and \(B\) the event \(p_{2}>p_{3}\). Are \(A\) and \(B\) independent?
8. (10 points) A random permutation is chosen uniformly from \([n]\). Are the events “\(1\) stays fixed” and “\(2\) stays fixed” independent?
9. (10 points) There are four coins \(C_{1},C_{2},C_{3},C_{4}\) with \(\Pr[ \text{Heads on }C_{i} ]=1/i\). A coin is chosen uniformly at random and tossed until the first Head appears. Given that the first Head occurred on the second toss, what is the probability that a chosen coin was \(C_{i}\) for each \(i\)?
10. (10 points) In a futuristic city, some people are robot spies that look just like humans. To counter this threat, the city uses an advanced facial recognition AI to detect robot spies. The AI has the following properties: If a person is a robot spy, the AI correctly identifies them \(98\%\) of the time. If a person is a normal citizen, the AI incorrectly flags them as a robot spy \(5\%\) of the time.

    One day, the AI scans a random citizen and flags them as a robot spy. After further investigation, it turns out that the actual probability of this person being a real robot spy is \(4\%\). What fraction of the city's population consists of robot spies?

11. (15 points) Consider a family with two children. Assume having a boy or girl is equally likely and independent, and being born on any day of the week is equally likely and independent.
    • (5 points) You are told “at least one is a boy”. What is the probability they are both boys?
    
    
    • (10 points) You are told “at least one is a boy born on a Tuesday”. What is the probability they are both boys?
    
    

12. (15 points) Two auditors split \(10\) companies equally (\(5\) each). Exactly two companies have violations. The first auditor detects any violation with probability \(0.8\), the second with probability \(0.9\).
    • (10 points) What is the probability both violating companies are identified?
    
    
    • (5 points) What is the probability that one violation is identified by the first auditor and the other by the second?
    
    

13. (15 points) Prove: if \(A,B,C\) are mutually independent, then \(A\) and \(B\cup C\) are independent.