Showing posts from 2022

Biased Best-of-K Rock Paper Scissors

The topic of today's blog post is an interesting twist on the classical rock paper scissors game. Alice and Bob agree to play rock paper scissors (best of 1).  If Alice loses, she loses gracefully and accepts defeat.  If Bob loses, he will insist on playing best of 3.  If he loses yet again, he will insist on playing best of 5, and so on and so forth. What is Alice's probability of winning this game if she is willing to agree to Bob's request $k$ times (best of up to $K=2k+1$)? Let's begin by working out the formula for $k=1$, which is perhaps the most realistic scenario.  How much does Alice give up by agreeing to Bob's request once?  The probability of Alice winning a given round is $1/2$, and same for Bob (a round is consists of a sequence of ties followed by exactly one non-tie).  Here are the possible sequence of events, along with their probabilities, with A/B denoting that Alice/Bob wins the given round respectively. B - 0.5 - Bob wins in 1 turn AA - 0.25 - A

Beat the Streak: Day Nine

In this blog post, I want to talk about why getting 80% success rate in beat the streak is so challenging.  I believe I identified a mathematical reason for this, which I am going to share in this blog post.   First, lets look at some simple statistics that hint that 80% success should not be out of reach.   In the table below, we are showing the percentage of games with a hit for the most successful batters in 2011-2019. batter % Games with Hit 2011 Jacoby Ellsbury 0.821656 2012 Derek Jeter 0.812121 2013 Michael Cuddyer 0.807692 2014 Jose Altuve 0.803797 2015 Dee Gordon 0.800000 2016 Mookie Betts 0.807453 2017 Ender Inciarte 0.775641 2018 Jose Altuve 0.786207 2019 DJ LeMahieu

Beat the Streak: Day Eight

In this blog post, we will explore three factors that influence the probability of correctly selecting a player to get a hit on a given day.  These are: 1. Individual batter strength, as measured by the proportion of plate appearances that resulted in a hit. 2. Team offensive strength, as measured by the average number of plate appearances per game by the batting team.   3. The position in the batting order. We plot the distribution of these statistics over (batter, year) pairs and (team, year) pairs.  The plots below reveal that the best batters get a hit in about 30% of plate appearances, and the strongest offensive teams average 39 plate appearances per game.  The tables below show the top-performing batters and teams: batter year Josh Hamilton 2010 0.326 Trea Turner 2016 0.324 Jose Altuve 2014 0.319