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?

Counting

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.