Hello everyone! I hope that you are all doing well during this crazy time. I have finally completed my MAT 380 seminar on The Birthday Paradox and Monty Hall Problem. It wasn’t easy, and I really struggled with LaTeX at first, but I think things really came together at the end. I really appreciate any advice, feedback, or tips that you have to offer. I also wanted to include a quick note before my abstract to thank all of those who were so kind as to help explain to me what seminars were all about, and assure me that my presentation topic was a good choice. I really appreciate it! Good luck to everyone with final exams approaching, and good luck to the seniors who are sadly leaving us!

### Abstract:

The study of probability is one of the most useful applications of math in the world around us. Yet, when we apply basic probability rules to real-world examples, the results can be pretty surprising and seem rather unlikely. Our expectations or perceptions of what goes on in the world around us may lead us to draw false conclusions about the likelihood of an event.

This is the case for both the Birthday Paradox and Monty Hall Problem (hence the words paradox and problem in their names). The Birthday Paradox demonstrates that the number of people you need in a room to have a 50% chance that two people share a birthday is much lower than most people would expect. The Monty Hall Problem is based off of the final round of the popular game show *Let’s Make a Deal*. This problem continues to stump mathematicians around the world, and is still disputed by some today. As my presentation will show, this problem boils down to a conditional probability problem, and recognizing this could actually increase the contestant’s odds of winning. My presentation

will demonstrate how these two problems can be easily solved with probability rules, despite their relatively surprising results when interpreted contextually.

Very interesting presentation topics. Either one could have stood alone as a solid presentation but to do both takes dedication so be proud of that. The slides are very well formatted and you couldn’t tell that this is your first time using Latex, the coding is just very nicely done. I always knew that you only needed a group of 23 people in order to have a shot at two to sharing a birthday but never knew why. So this is helpful learning that this is just using a simple probability function. Also, as a slightly above average fan of the show Let’s Make a Deal, I’m very happy to see this show incorporated into sem. Nice work.

Great job with the presentation, it was formatted very well. Additionally, as someone who enjoys probabilities, learning about the background to the Birthday Paradox with interesting. I thought that extending the probability theory into Monty Hall problem was also a great idea, as you can see how they flow together in the presentation.

Finally, the conclusion slide was great, and showed exceptional thinking in how these concepts could be applied further! Maybe you could do something similar for your 381/480 seminar.

This was a great presentation. It was well organized and easily understood. I like this topic, it made me think about the show Deal or No Deal. They used to ask the contestant if they wanted to switch briefcases at the end. This would be the same thing as the prized-door example, would it not? They would have a better chance if they switched briefcases. I really like this concept and its easily understood for anyone reading about it. When you first started talking to me about it I was interested because it was crazy to think that you only need 23 people to have a 50% chance that two people in the room have the same birthday. You did a great job with the presentation.

Thank you so much for all of your kind comments! It really means a lot! As far as your question as to whether or not the final two cases on Deal or No Deal would take a similar form as the Monty Hall Problem, I’m not sure that it would work out. Some of the conditions required for the Monty Hall Problem are violated mainly because of the differences in the structures of the two games. For Deal or No Deal, the contestant is the one selecting doors, and they have no knowledge of where the prizes are. For the Monty Hall Problem, the host knew exactly which door had the prize, and had to avoid opening that door, and it was because of this that the contestant’s odds of winning were greater when they switched doors. One of the articles I used in my presentation went through a scenario with 100 doors instead of 3. The contestant only picks one just like before, but then the host opens 98 of the remaining doors. The contestant’s original door only had a 1/100 chance of containing the prize. However, the host cannot open the winning door, so he is essentially filtering the doors for the contestant. As far as I know then, switching cases would not increase the contestant’s odds of winning on Deal or No Deal, and their odds of winning the higher prize would be 50-50 regardless. However, I am not 100% sure about this. I will definitely take a look into this and add in a new comment if I find something different. Thanks so much for the question! Also, thank you so very much for helping me understand how seminars work! I was completely lost until you were kind enough to tell me what LaTeX even was!

I really enjoyed this presentation. Although I thoroughly enjoy regression techniques and such, I was never able to get a handle on probability. Maybe it’s intimidating for me, I don’t know. But you did a really nice job, and it was quite ambitious for a 380 talk! Last year, a former Canisius math professor came and gave a talk on the birthday paradox, so you’re in good company. Great job!

That was a great presentation. I really enjoy your sides and the topic was interesting. It was well organized and it was easy to follow. Although I am not that good at probability stuff, I can tell how much afford you have put it on your slides. Thank you for sharing your information on the Birthday Paradox and Monty Hall Problem.

The Monty Hall problems has plagued my mind for years. i could never quite make sense of it, until i saw your demonstration and visualizations. I finally feel like i have closure on this problem. The formulaic way you went about the demonstration was very helpful. Excellent work!

Your presentation clearly exemplifies the hard work and passion you have for probability! I have heard people mention the Monty Hall problem before, but have never explored the probability behind game shows. Thank you for taking the time to make a video as it was a really cool way to demonstrate this problem. I also appreciate how you mentioned Deal or No Deal because I have always wondered why the banker creates offers the way they do. There are numerous games in The Price is Right that I feel may benefit from conditional probability as well (although some are skill/chance games). Well done especially for a 380!

Great job on your presentation. I really admired that you chose two topics, even though one would have been enough for a 380 talk. It clearly shows that you are willing to put in the time and effort. Both of your topics can be incorporated into real life scenarios, so it was interesting reading about the math behind it. I really appreciate you including the pictures of the doors for the Monty Hall problem because it made it easy for me to wrap my head around the scenario. I was confusing myself trying to sort and solve the numbers in my head.

I love weird math stuff like this. There are some really famous math problems like the monty hall problem that are just totally the opposite of what your instincts are telling you. As often as I hear the math and the proofs and explanations for them, I can never wrap my head around them. You put a lot of work into your presentation and picked a cool topic. Great job!

Very neat problems and very manageable maths! Thank you for your contribution, combinatorics and probability are very interesting fields of statistics and both of these problems play into them well. I would have absolutely assumed there was a 50/50 chance after one of the losing doors were opened, but your explanation was very enticing!

Good work!

Very interesting presentation idea. Both of your examples are also very curious. About Birthday Paradox, I was really surprised after knowing the result that just 23 needs for 50% chance。Although I have never asked 23 people when your birthday is, I cannot believe this result!! I think these problems are not only funny and interesting problems but also useful way to real world. If I assumed I gonna be a host of a matchmaking party of 23 people and also supposed 365 personal characteristics exist, I might be able to guarantee that 2 participants can be a couple who share the complete same with 50% in a group.(I don’t think it is a good example but it can be a good solution in a way, I guess…) Anyway, I like your presentation. Great job！