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:...