Ball in Hand vs. Two Visits ??

In English pool, there are two distinct sets of rules governing fouls: the official regulations and the commonly referred to "Supreme rules." According to the official rules, when a player commits a foul, the opponent gains the advantage of placing the cue ball anywhere on the table and making their shot. In contrast, under the more casual bar rules, typically embraced by non-professional players, fouls result in a different penalty: the offending player grants their opponent not one but two successive visits to the table but without moving the cue ball. This means that even if the opponent misses their shot during the first visit, they retain the opportunity to continue playing until they miss again on their second visit.

As someone who regularly indulges in the game of pool, I often find myself adapting to these varying rule sets, depending on the preferences or familiarity of my opponents. One fateful day, while engaged in a friendly match with my friends Mouad A. and Saad HARISS , Mouad posed a thought-provoking question: "Can you statistically prove which option is more advantageous when your opponent commits a foul, i.e., having the cue ball in hand or enjoying two consecutive visits?"

This question sparked our curiosity, and it serves as the foundation for our exploration here. In the following discussion, I aim to shed light on the comparative merits of these two penalty options by considering various factors, such as the skill level of the players and the number of remaining balls on the table. I will adopt a comprehensive approach to address this issue and provide a logical resolution to our intriguing query.

Summary:

• Introduction
• Discussion
• Equations & Graphs
• Conclusion

Introduction:

Initially, the three of us faced some difficulty in approaching this challenge. Each of us had our own unique perspectives, and after much discussion, we decided that each one should articulate their approach through a LinkedIn article. In this piece, I will share my perspective.

To prove which rule is statistically better, we first need to describe playing pool in a way that involves math. I'm not a math expert, but I did take a Probability and Statistics course during my engineering studies. My initial idea was to apply what I learned from that course to tackle this problem.

But before we can provide a definite answer, we need to address some key questions:

• What does it mean to have an advantage?
• What factors should we consider that might affect our answer, like the skill level of the players, the quality of the pool table, the playing conditions, and the quality of the cues?
• How can we represent this problem using equations and statistics?

Discussion

I began by providing some definitions for the key terms in the question. For instance, I defined "advantageous" as the ability to pocket more balls. This means that if, in a certain situation, having two consecutive visits allows you to pocket more balls compared to having a ball in hand, then I would consider the bar rules as more advantageous. Keep in mind that this discussion will focus on the scenario where putting more balls after your opponent commits a foul is our primary concern. I understand that merely pocketing more balls doesn't guarantee a win, but it certainly increases your chances of winning.

The second aspect I aimed to clarify was the factors that might favor one rule over the other. Take, for example, a highly skilled player who can clear the table in a single visit. For such a player, being granted two consecutive visits is akin to saying, "Sure, go ahead, I want you to win." This brings us to the player's skill level, which is a crucial factor we must consider in our discussion. As for other factors like the quality of the pool table, I couldn't definitively determine how they might influence our discussion. For the purposes of our analysis, I will assume that the difference between the two rules remains consistent in the same playing environment, thus not significantly impacting our equations later on.

Equations & Graphs

To begin translating our thoughts into equations, I needed a way to represent a player's skill level as a numerical value, as that's the primary variable I'll be examining. To accomplish this, I had to answer a fundamental question: "What sets apart a good pool player from a less skilled one?" The answer is quite evident – a good player consistently pockets more balls than a less skilled player. In simpler terms, a less skilled player is more likely to miss their shots, and when we discuss likelihood, we are essentially talking about probability. So, if we randomly select any pool position, the question becomes: Who is more likely to pocket the ball? Undoubtedly, a more skilled player. This led me to represent a player's level by their probability of pocketing a ball in any random position. By this definition, a terrible player would have a skill level of 0, while an exceptionally skilled player would have a skill level of 1. It's important to acknowledge that neither extreme exists in reality because no player can miss every shot or make every shot. Hence, I introduced a variable, denoted as "p," to represent skill level.

Upon addressing this question, the next inquiry that naturally arises is: How do we quantify the difference between having a ball that can be placed anywhere on the table and having two consecutive visits?

• Ball in hand: When a player has the cue ball in hand, they can position it anywhere on the table and take their shot. This significantly increases the probability of pocketing the first ball, but the outcome still depends on the player's skill level. If we call this probability "Pbih" (probability of pocketing a ball in hand), we can express it with the equation (for a specific player): p < Pbih < 1. However, the probabilities for subsequent shots remain at the player's skill level, "p."
• Two consecutive visits: If an opponent commits an error, and we have two consecutive visits without disturbing the positions of the balls, it means the cue ball and target balls can be located anywhere. Consequently, the probability of pocketing one of the target balls remains at the player's skill level, "p."

Scaling for N balls

Now, let's move on to the next step, where we need to scale the probability to accommodate N balls. This is an important aspect to consider, albeit with a general approach. It's important to note that there may be specific scenarios where these calculations might not be entirely accurate, but statistically, we can reasonably expect them to hold true.

To start, let's focus on the case where we aim to pocket N balls in succession, beginning with having the cue ball in hand:

Well if putting a ball has a probability of p then the probability of putting two balls in a row is simply putting a ball and a second ball, which is equal to p * p. which mean putting N balls in a row is simply p to the power of N. However, when you have a ball in hand, the probability of the first ball being pocketed is denoted as "Pbih." This implies that the probability of putting N balls in a row with the first ball hand is:

and because we don't know Pbih for sure let's frame the above probability,

Let's consider the scenario of putting N balls in a row, using two visits:

As previously mentioned, in this option, we still have the same probability for each ball, but we have the opportunity to miss once and continue playing. To calculate P2v (the probability of putting N balls in 2 visits) in this configuration, we need to account for various scenarios:

1. We could put all N balls in our first visit, which is simply represented as p to the power of N.
2. Alternatively, we might put 1 ball in one visit and the remaining N-1 balls in the other visit. This scenario involves putting one ball, missing the second ball, and then putting the N-1 balls in a row. The equation for this is p (1 - p) p to the power of N - 1. It's important to note that the order in which we miss a ball doesn't matter; the probability remains the same at p to the power of N * (1 - p).

Indeed, as observed, we have a total of n+1 scenarios to consider. The first scenario entails not missing any ball, while the other n scenarios involve missing one ball in each. In each of these n+1 scenarios, we aim to put N balls within the confines of our two visits.

This leads us to the following equation, where the probability P2v is the sum of the probabilities for all these scenarios:

Now, we need to compare the following two equations to determine which scenario is more advantageous and to arrive at our final answer:

Since we don't have a specific value for Pbih, we'll represent it as an interval or a range, demonstrating the range of values between p and 1.

Now, let's transition to GeoGebra to graph these two functions and gain a visual understanding of the disparity between the two equations or pool rules.

Case where we only need to put one single ball:

In the graph, the red curve represents P2v, and the blue area corresponds to the ball in hand area. The x-axis represents the player's skill level, while the y-axis indicates the likelihood of putting N balls.

From the graph, it's evident that the curve falls within the blue area, making it challenging to definitively determine which option is superior when only one ball remains. However, an interesting observation is that as the player's skill level increases, the choice becomes less critical.

Now, let's explore the impact of increasing the number of balls and see how it alters the situation!

Case where N = 4:

Absolutely, there's a significant difference! As we increased the number of balls to put in a row, the blue area became considerably smaller, and the red curve now sits entirely above the area. This implies that having two visits is indeed far more advantageous than having a ball in hand. While it's true that a highly skilled pool player will benefit even more from two visits, it's worth noting that even for a mid-level player (p=0.5), choosing two visits over a ball in hand doubles their chances of pocketing four balls.

This difference becomes even more pronounced as we can see in the next image, with N=8 (the maximum number of balls you can put in English Pool to win).

Conclusion:

Do not use bar rules when playing against a more skilled opponent, as this significantly increases the likelihood of losing the game if you commit any fouls. ??