## Practice Problems (Introduction to Probability Theory)

A few practice problems related to Probability theory are given here. These problems are adapted from the textbook: Probability and Random Processes by Scott Miller 2nd Edition. These problems cover the following topics: Experiments, Sample spaces, events, Axioms of Probability, Assigning Probability, Joint and Conditional Probability, Independence, Baye’s Theorem, Total Law of Probability, Discrete Random variables. Problem-1: A roulette wheel consists of 38 numbers (18 are red, 18 are black, and 2 are green). Assume that with each spin of the wheel, each number is equally likely to appear. (a) What is the probability of a gambler winning if he...

## Baye’s Rule

In Probability theory, we came across some problems in which it is difficult to compute conditional probability directly. To explain this let us consider two events A and B, then their joint probability is given as: $$\begin{eqnarray}P(A,B) = P(A|B)P(B) \label{eqA}\end{eqnarray}$$ and we know that: $$\begin{eqnarray}P(B|A) = \frac{P(A,B)}{P(A)} \label{eqB}\end{eqnarray}$$ then using \eqref{eqA} and \eqref{eqB} we get: $$\begin{eqnarray}P(B|A) = \frac{P(A|B)P(B)}{P(A)} \label{bayes}\end{eqnarray}$$ Equation \eqref{bayes} is called Baye’s Rule and is useful for calculating certain conditional probabilities as in many problems it is difficult to compute P(B|A) directly, hence we use Baye’s Rule to solve such problems. A few examples are given below:...