Probstat/week2 practice 1

This is part of probstat

Conditional probability review

1. Consider a sample space $S=\{0,1,2,3,...,41\}$. We randomly pick an integer from $S$.

• 1.1 What is the probability that a random number from $S$ is divisible by 2 given that it is divisible by 3.
• 1.2 Are events "an integer divisible by 2 is picked" and "an integer divisible by 3 is picked" independent?
• 1.3 What is the probability that a random number from $S$ is divisible by 4 given that it is divisible by 3.
• 1.4 Are events "an integer divisible by 4 is picked" and "an integer divisible by 3 is picked" independent?

2. The following table shows probabilities of events related to people having lung cancer and smoking cigarette. (Numbers are made up). That is, if an event $S$ is an event that a random person is smoking and $C$ is an event that a random person has lung cancer, the number in the top-left cell is $P(S\cap C)$, while the number in the bottom-right cell is $P(S^{c}\cap C^{c})$.

	Smoking	Not smoking
Cancer	0.12	0.07
No cancer	0.03	0.78

• 2.1 What is the probability that a person has lung cancer if that person smokes?
• 2.2 What is the probability that a person does not have lung cancer if that person does not smokes?
• 2.3 What is the probability that a person smokes if that person has lung cancer?
• 2.4 A stupid cancer predictor works like this: it asks a person if that person smokes, if the person answers yes, the predictor predicts that the person has lung cancer. Otherwise, the predictor predicts that the person does not have lung cancer. What is the probability that the predictor is wrong? (More precisely, suppose that we pick a random person, what is the probability that the predictor gives incorrect prediction?)

Counting

You may find it helpful to watch Counting Part 2.1, 2.2, 2.3 first.

1. We have a bag with $n$ different objects. We randomly choose $k$ objects from the bag, one by one, without replacement. I.e., we choose the first object randomly, keep it, then we choose the second object randomly, keep it, and so on. How many ways can we choose $k$ objects?

From the previous question, if we let $k=n$, i.e., we choose all objects, for a certain way of choosing, we obtain on ordered arrangement of $n$ objects. This arrangement is called a permutation.

2. How many permutations of $n$ objects are there?

3. (From FCP, ex 3b) In a running competition, there are 2 groups of players: blue group and green group; each group has 4 players. Assume that there are no ties.

• 3.1 How many different final rankings are possible?
• 3.2 How many different final rankings where the players from the green group take the first 4 positions?

4. (From FCP, ex 3c) There are 4 different math books, 3 different history books, 2 different chemistry books, and 1 literature book. We want to put these books in a shelf so that books from the same subject are together on the shelf. How many ways to do that?

5. How many different letter arrangements can be formed from the word PROBABILITY?

6. There are 5 female and 4 male students in a class. We would like to pick 4 students as representatives.

• 6.1 How many different sets of representatives are possible?
• 6.2 If we want a set of representatives to consists of 2 female and 2 male students, how many different sets of representatives are possible?
• 6.3 If we want a set of representatives to consists of students from two different genders, how many different sets of representatives are possible?

7. We would like to assign ID's to students' labtops. An ID is a 4-character string consisting of alphabets A, B, C, D, E, and F, that satisfies the conditions that (1) each alphabet appears at most once and (2) the string only contains at most one vowel. How many different ID's are there?

8. (From FCP, Ch1. Problems 2) Alice, Bob, Cathy, and Dave is forming a band that plays 4 instruments. If each member can play any instruments, how many different arrangements are possible? If Alice and Bob can play any instruments but Cathy and Dave can each play only piano and drums, how many different arrangements are possible?