Probstat/homework 1

This is part of probstat.

In these problems, please hand in your paper solutions. Don't just state the answers, but also show your work.

Basic problems

1. Work on problems 1, 3, 5, 7, 9, and 10 in Counting section in week2 practice 1.

2. (From FCP, Ch 2, Problem 13.) We have two bags. Bag A contains 4 red and 3 black balls. Bag B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be of the same color?

3. (From SPS, Solved Exercise 1.15) Consider the following experiment. We have two boxes. Box 1 contains 3 red and 2 blue marbles while Box 2 contains 2 red and 8 blue marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box 1, otherwise it is chosen from Box 2.

• 3.1 What is the probability that a red marble is chosen?
• 3.2 If we has chosen a red marble, what is the probability that the marble is chosen from Box 1?

4. (From SPS, Solved Exercise 1.27) In how many ways can 10 different objects be split into two groups containing 4 and 6 objects, respectively?

5. (From SPS, 1.53) Find the probability of drawing 3 aces at random from a deck of 52 cards, if the cards are

• 5.1 replaced.
• 5.2 not replaced.

6. After you got hit by the rain, you will get sick with probability 0.4. Given that you are sick, you will get a headache with probability 0.5. However, if you are not sick, you can still get a headache with probability 0.1. Assume that you got hit by the rain yesterday.

• 6.1 What is the probability that you are sick and get a headache?
• 6.2 What is the probability that you are not sick but get a headache?
• 6.3 What is the probability that you get a headache?
• 6.4 Given that you get a headache, what is the probability that you are sick?

7. There are 4 runners: A, B, C, and D in a running competition. Since their ability are very close, any rankings from the competition are equally likely.

• 7.1 How many possible rankings are there?
• 7.2 What is the probability that A gets a better rank than B?
• 7.3 If we know that C beats D, what is the probability that C gets the first or the second places?
• 7.4 If we know that C beats D, what is the probability that D gets the first or the second places?

Theoretical problems

1. Use the axioms of probability to prove that $P(\emptyset )=0$.

2. For any events A and B, show that if A and B are independent, then A and B^C are also independent.

3. Consider the following experiment: we toss a coin that turns up head with probability p for n times, independently. Prove that the probability that we get exactly k heads is

${n \choose k}p^{k}(1-p)^{n-k}$.

4. We toss a coin that turns up Head with probability p for many rounds independently. What is the probability that you get Head in the k-th round and get Tail in all other rounds 1, 2, ... , k-1?

5. We are tossing N different balls into N bins. Each ball is equally likely to land in any bins. What's the probability that there is no empty bins? (I.e., empty bins are those with no balls landed on.)

Challenging problems

1. There are n people (n <= 365). Assume that there are 365 days in a year and a person's birthday is equally likely to be on any day in a year and birthdays are independent.

• 1.1 What is the probability that these n people have different birthdays? (You do not have to solve for the closed-form formula.)
• 1.2 Write a program that finds the minimum value of n such that the probability above is less than 1/2.

2. A biased coin tuns up head with probability 0.7. The probability that a random coin is biased is 0.1. Let event A_k be the event that you toss the coin k times and the coin turns up all heads. Let event B be the event that the coin is biased. Compute the following conditional probability: P(B), P(B|A₁), P(B|A₂), P(B|A₃), P(B|A₄), and P(B|A₅). Plot the probabilities in a chart where the x-axis shows P(B|A_i), where i ranges from 0 to 5.

3. Can you define a random experiment together with 3 events A, B, and C, such that these events are pair-wise independent (i.e., A and B are independent, A and C are independent, and B and C are independent), but the 3 events are not mutually independent?