The Monty Hall problem is an interesting puzzle loosely based on an American TV game show Let’s Make a Deal hosted by Monty Hall. While the puzzle looked simple, it perplexed some of the brightest mathematical minds in the United States, including the great Paul Erdös who was one of the most prolific mathematicians of the 20th century. This continues to be the case today. I looked upon a number of references to find the source of confusion in the Monty Hall problem but failed. All I found was different solutions. Therefore, I built one myself with the usual from

Continue reading# Tag: Probability

## The Coin Toss Puzzle and the Simplest Possible Solution

Recently, I wrote an article on why the Monty Hall problem has perplexed so many brilliant minds where I showed that it was a corner case between 1 open and 1 closed door, while the intuitive but wrong answer is close to the probability curve of 1 open door. Now a coin toss puzzle has appeared on Twitter that has gone viral as it goes against our common intuition of probability and random sequences (such as a series of coin tosses). The puzzle goes as follows. The Problem Flip a fair coin 100 times—it gives a sequence of heads (H)

Continue reading## Maximum Likelihood Estimation of Clock Offset

When I started my PhD, one of the first papers I read was On Maximum Likelihood Estimation of Clock Offset by Daniel Jeske [1] from University of California, Riverside. It eventually set the direction of my future research and ultimately my PhD dissertation. I found this paper quite interesting as it talked about the estimation of clock phase offset. Later I went on to explore what was missing here (the clock frequency offset) and more. Keep in mind that carrier phase estimation is a different problem that has already been discussed in the past here, here and here. Most of

Continue reading## Diversity in Wireless Communication Systems

Diversity is one of those few ideas that are extremely dumb and extremely clever at the same time. It can be explained in one sentence as well as in a whole book. The basic idea, nevertheless, is quite simple. What is Diversity? Consider the following two different cases. Many phenomena in the world need a series of outcomes to succeed. For instance, for life to exist in the cold and dark universe, we need a star to provide energy as heat. A planet is also required as a home. Furthermore, this planet must reside in the Goldilocks Zone of that

Continue reading## Computing Error Rates

Having built a simple digital communication system, it is necessary to know how to measure its performance. As the names say, Symbol Error Rate (SER) and Bit Error Rate (BER) are the probabilities of receiving a symbol and bit in error, respectively. SER and BER can be approximated through simulating a complete digital communication system involving a large number of bits and comparing the ratio of symbols or bits received in error to the total number of bits. Hence, \begin{equation}\label{eqCommSystemSER} \text{SER} = \frac{\text{No. of symbols in error}}{\text{Total no. of transmitted symbols}} \end{equation} and \begin{equation}\label{eqCommSystemBER} \text{BER} = \frac{\text{No. of bits in

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