# Bacteria Allocation

• Objective
• Level
• Prerequisites
• Platform
• Instructor's Notes
• Credits
•
Objective: To develop a procedure for randomly allocating bacteria to a fixed number of droplets from a total of 100 droplets using a probability experiment employing the Monte Carlo technique. (For a general description of the Monte Carlo technique go to the Elementary Probability Demo Collection.)

Level: High school or precalculus, or probability courses for math majors.

Prerequisites: Basic probability concepts; knowledge of equally likely chance and the chance of winning a lottery.

Platform: A calculator or computer which has a function that generates random numbers. There is an accompanying JAVA applet which performs simulations discussed in this demo: javabacteriasimulation.

Instructor's Notes:

This demo uses the Monte Carlo technique to simulate a count of bacteria that are present as a result of a certain sampling process. The process extracts a fixed number of droplets N from a sample of contaminated liquid in which bacteria are randomly distributed. It is observed that (on the average) only a certain percent K of the droplets contain the bacteria. We want to approximate (on the average) the number of bacteria that are present in the droplets.

A scientist collects N droplets from a sample of a liquid containing bacteria. When the scientist studied the droplets, she found that only K% of the N droplets collected contained bacteria. This process was repeated a number of times and it was found that on the average only K% of the droplets were contaminated with bacteria, but the total number of bacteria from those droplets varied. To approximate the average number of bacteria contained in the contaminated droplets of this particular liquid the following simulation was devised.

For experimental purposes set N = 100 and K = 50. Imagine that the 100 droplets are arranged in a set of containers that form a square, 10 containers by 10 containers. Each of the containers is assigned a number from 1 to 100. (See the numbers in black in the lower left corner of the containers in Figure 1.) We randomly generate integers from 1 to 100. When a particular container's number is generated we add 1 "bacteria" to the container. We repeat this process until 50 containers contain at least one bacteria and we keep a running total of the number of bacteria that are allocated to the containers. In Figure 1 below we show a result of this simulation. In this particular experiment 73 bacteria were needed to have 50 of the 100 droplets contain at least one bacteria.

Figure 1.

By repeating the procedure a large number of times with N = 100 and K = 50 we can obtain a good estimate of the average number of bacteria that occur in the 50 droplets.  The MATLAB routine bacteriasim.ZIP (click to download) implements the strategy outlined above. (Figure 1 and the animation below were obtained from this routine.)  We have also developed a JAVA applet for this demo which has the same functionality :  javabacteriasimulation. (The displays of the applet may vary slightly because of the browser you use and the screen resolution of your monitor.) We repeated the simulation 10 times and the results are displayed in the following table.

The theoretical solution of this Monte Carlo process is 69. The average of the 10 experiments reported in the preceding table is 70.2,  a close approximation. We expect the average number of bacteria in the 50 droplets to approach 69, when a large number of experiments are performed.

The animation below shows a typical simulation of the random allocation process for N = 100 and K = 50.

This simulation could be modified to perform other experiments. The MATLAB routine bacteriasim (cited above) and the applet  javabacteriasimulation lets the user specify the number of droplets to contaminate from the 100 collected. Selecting a different number of droplets to contaminate will alter the (average) number of bacteria expected in those droplets. Table 1 below shows the average number of bacteria needed to contaminate K% of N = 100 droplets from a set of 10 experiments performed for each value of K.

The averages for a set of experiments you perform like those reported in Table1 will vary somewhat. In order to obtained very accurate estimates of theoretical probabilities involved quite a large number of experiments should be performed. To see details of the experiments summarized in Table 1 click on bacdataset.

Credits:  This demo was adapted from

W.E. Haigh, "Using Microcomputers to Solve Probability Problems",  Mathematics Teacher, v78, No.2, 1985, pp124-126

We also appreciate the cooperation of

Professor W.E. Haigh
Department of Mathematics
Northern State University

in its development.

We also acknowledge contributions by Un Jung Sin , student at Temple University. The MATLAB routine bacteriasim was written by David R. Hill. The Java applet was developed by Philip Nicastro and Michael Shelmet, both students at Temple University.

DRH 5/24/01    Last updated 1/25/2005

Since 6/14/01