Ping pong (or table tennis) is a game of both odds and luck. Most games I have seen have been rather one sided — it is really rare that two players are really at the same level. Pretty much every game I see at our ping pong (that is, table tennis) table at work is rather unbalanced. Even if two opponents had the same chance at winning any given point, differences in their excitement levels across multiple points can impact the next points — people can be “on a roll” or “choke” and temporarily increase or decrease their change of winning the next point.
This week’s puzzle is about two players, one who is completely constant (let’s call him Troy), and one who occasionally gets “on a roll” and “chokes” (let’s call her Katharine.) For purposes of this puzzle, on any given day Katharine will be prone to either being “on a roll” or “choking,” but not both. Here is how we will define each:
Choking – If Katharine wins two points in a row at her “standard” point probability, her odds for winning the next point will decrease by 25 percentage points. Furthermore, her odds will continue to decrease by an additional 25 percentage points for each additional consecutive point. For example, if she has a standard 60% chance of winning any given point, after winning two points in a row her odds for the next point will drop to 35%. If she wins that third point, her odds for the fourth point in a row fall to 10%. Clearly, it is not possible for her to win five consecutive points on days when she is prone to choking. After losing the point by choking, Katharine immediately returns to her standard odds for the very next point.
On a Roll – If Katharine is prone to being “on a roll” for a given day, winning two points in a row improves her odds for winning the next point by 25 percentage points. She will continue to keep that 25 percentage point advantage until she loses a point or the game ends — at which point she reverts back to her “standard” chance of winning the next point. For example, if Katharine has a standard 55% change of winning any given point, after winning two points in a row her odds will increase to 80% until she either loses a point or the game ends, and which point she immediate returns to her standard odds for the very next point.
Clearly if Katharine’s standard odds are even with Troy’s (i.e., 50% / 50%), Katharine will tend to win over time on days that she is on a roll, and will tend to lose over time on days that she is prone to choking. However, looking back through the years at their game records, they both won and lost the exact number of games.
If their records are indeed accurate, what is Katharine’s “standard” point probability for days when she is prone to being “on a roll?” Additionally, what is Katharine’s “standard” point probability for days when she is prone to choking?
NOTE: Troy is completely constant in his odds for winning a point throughout any given day, but changes to be the exact counterpart of Katharine’s for each day. A standard game of ping pong (table tennis) is 21 points, with the winner having to win by two points.