Chocolate Chip Allocation

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
    Objective: To develop a procedure for randomly allocating chocolate chips to cookies 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.

    Instructor's Notes:

    This demo is an allocation simulation that uses the Monte Carlo technique. That is, we have a certain resource that is to be distributed in a random manner. We want to approximate the probability of certain outcomes. Here we consider randomly distributing chocolate chips to cookies.

    A cookie manufacturer bakes a batch of 100 cookies and decides to randomly allocate N > 0 chocolate chips to the batch. For advertising and quality control, the manufacturer wants to estimate the (average) number of cookies in a batch that will have no chocolate chips.

    A way to simulate the allocation of the chocolate chips to the cookies is to imagine that the batch of 100 cookies is arranged in a table consisting of 10 rows and 10 columns. Using this model for the batch, each cookie can be located by its row-column position. We generate pairs of random numbers (x, y) where both x and y are integer values from 1 to 10. Each time a particular (x, y) pair is generated we add 1 to the table entry at that location.

    After the N chips have been allocated, we count the number of cookies with no chips. This provides an estimate of the average number of cookies containing no chips in a batch of 100 cookies to which N chips are allocated. Figure 1 shows one such allocation where the red zeros indicate the cookies with no chips. In this simulation we used N = 150. There were 20 cookies with no chips and the largest number of chips allocated to one cookie is 4.

                                                                Figure 1.

    By repeating the procedure a large number of times keeping N fixed at 150 we can obtain a good estimate of the average number of cookies with no chips for a batch of 100 cookies. The MATLAB routine chipsim.ZIP (click to download) implements the strategy outlined above. (Figure 1 and the animation below were obtained from this routine.) We repeated the simulation 10 times using a batch of 100 cookies with an allocation of N = 150 chips. The results are displayed in the following table. 

    The average number of cookies with no chips from this set of 10 experiments is 22.4. We stress that this is only an estimate of the average number of cookies contain no chips when N = 150 chips are randomly allocated to the 100 cookies. A larger set of experiments may be required to accurately estimate the "on average" number of cookies with no chips for arbitrary choices of the number of chips N.

    The animation below shows a typical simulation of the random allocation process for N = 150.

    This simulation can be modified in a variety of ways. The MATLAB routine chipsim (cited above) lets the user specify the number of chips N to allocate to the 100 cookies. Selecting a different number of chips to allocate will change the estimates of the number of the 100 cookies that, on average, contain no chips. Another way to modify the simulation is to change the number of cookies in a batch. In general, simulations of this type provide an estimate of the probability of selecting a cookie with no chips from a batch of C cookies to which N chips have been randomly allocated. The estimate is computed by the ratio
    In Table 1 we summarize a set of experiments for various choices of N, the number of chips to be allocated to C = 100 cookies. For each value of N 10 experiments were performed and the indicated averages computed.

                                                Table 1.

    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 dataset.

    In advanced courses,  theoretical  techniques can be used to compute the such probabilities. For example, the following table gives the theoretical probabilities as indicated for a batch of 100 cookies randomly allocated N = 150 chips.


    Credits:  This demo was adapted from  

    W.E. Haigh, "Using Microcomputers to Solve Probability Problems",  Mathematics Teacher, v78, No.2, 1985, pp124-126
    and we 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 chipsim was written by David R. Hill.

    DRH 5/23/01

    Since 6/14/01