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 probability of people having 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

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?