 Issue: 3/05

Plumbing designers are often called upon to design medical gas piping systems, including oxygen, nitrous oxide, nitrogen, carbon dioxide, medical vacuum, WAGD (Waste Anesthetic Gas Disposal) and medical compressed air. As with any gas piping system, in order to determine pipe sizes accurately, we must first determine the maximum anticipated flow. Then, once the flow has been determined, pipe sizes may be selected using standard tables or by direct calculation using various gas-flow formulae.

However, calculating medical gas flows and pipe sizes, especially for large facilities such as hospitals, can be a rather complicated process. To determine flow, designers have traditionally counted the number of outlets/inlets served by a section of piping without regard to the location of those outlets/inlets, applied a "use factor" Jacob Bernoulli (1654-1705) discovered many of the fundamental principles of probability, including the “Theory of Large Numbers.” These are discussed in the work Ars Conjectandi, published in Basel, Switzerland, in 1713, eight years after his death.

## Medical Gas Demand and Probability

Whenever scientists, engineers and other researchers are called on to calculate certain quantitative information, they often turn to the principles of probability. Probability is a branch of mathematics that is used to determine what is likely to occur (not necessarily what will occur) in a given event or sequence of events. Swiss mathematician Jacob Bernoulli (1654-1705) is credited with the discovery of many of the fundamental principles of probability, including the binomial equation used to generate probability distribution curves, as well as the "Theory of Large Numbers." Equation 1
First, let's look at a binomial probability distribution. A binomial probability distribution is used to determine the probability of a certain number of favorable results for a given number of trials with dichotomous outcomes (i.e., a trial with only two possible mutually exclusive outcomes-also referred to as a Bernoulli trial). For example, we may use it to determine the chances of getting eight heads from 15 tosses of a coin, or the chances of getting four twos in 10 tosses of a die, etc.

The binomial formula (Equation 1) is shown at right.

Equation 1

Where:

P(x) = Probability of getting x positive outcomes

n = number of trials

p = probability of a single trial being successful (e.g., p = 0.5 for Probability of getting a head on a single toss of a coin and p = 1/6 for the probability of getting a two on a single toss of a die, etc.)

x = number of positive outcomes

Equation 1-proven by Jacob Bernoulli in 1685-is used to generate the well-known binomial probability distribution curve, the familiar "bell-shaped" Equation 2
To predict a result with a particular confidence level, we multiply the standard deviation mentioned above by a "z" Figure 1
For example, if we want to determine the maximum number of heads we could expect in 10 flips of a coin with a 97.5% confidence level, then we would choose 5 + 1.96 x ((10 x 0.5 x (1-0.5))1/2 = 8.099 heads. In other words, to be 97.5% "safe," Figure 2
We may express Equation 2 another way by dividing each side of the equation by "n." Equation 3
As "n" Figure 3
As another example, let's say that statistics indicate that for a very large population of youngsters, one out of every five will buy an ice cream cone at a carnival. We are planning on selling ice cream at an upcoming carnival. If we know that about 150 kids of this age bracket are going to attend, then how many ice cream cones should we have on hand? Should we figure 20% of 150? Not necessarily. Equation 3 indicates that the percentage depends on the confidence level we desire, the chance that an individual child will buy one, and the number of trials (youngsters in this age bracket we expect to attend). The larger the number, the closer to 20% we should figure; the fewer the number we expect to attend, the higher the percentage that we'd need to figure to be assured of having enough supplies. Example 1. Use factors for medical vacuum in ICU areas. Note that the use factor approaches the value of “p ” as the number of beds increases. Also note that the graph indicates a 125% use factor for one bed. This is due to the nature of the binomial formula. We would obviously use a 100% use factor for any values above this number.

## Theory of Large Numbers: A Basis for Determining Probabilities

As was pointed out in the examples above, in order to determine the number of positive results that one needs to consider with a stated confidence level, we must not only know the number "n" Equation 4
However, what about other cases, such as the probability of a youngster buying an ice cream cone at a carnival? In order to determine such probabilities, we must turn to the "Theory of Large Numbers" Example 2. Use factors for oxygen in medical/surgery patient rooms.
According to the TLN, we must sample a large number of the total population in order to determine what the true probability is. In the previous example, it was stated that the probability that a youngster would buy an ice cream cone at a carnival was one in five (20%). This statement could only be stated confidently if a very large number of youngsters were polled. Smaller samples could easily give us misleading results.