### Stock & Flow Models

### Randomness and Monte Carlo Simulation

Consider this classic example of a causal loop diagram. Explain each of the plus or minus signs. What do the R and B loops indicate?

Consider this diagram. Explain what is going on .

Consider this stock and flow diagram showing relationship between a predator (lynx) and prey (rabbit) population. There is a "causal" arrow pointing from the stock of rabbits to the stock of lynx's and vice versa. You note that this is not quite correct. The lynx population size actually affects the death rate of rabbits (the more lynx's the more rabbits get eaten) and the rabbit population affects the death rate of lynx's (the fewer rabbits, the more lynx's starve to death). Re-route these two arrows to show this correction.

Consider a dropout prevention program at a high school in Oakland. It has asked you to put together a stock and flow model showing the basics of how kids "flow" through the system. They give you the following information. Kids enter the first year of high school, ninth grade, from 4 feeder schools. The high school principal tracks students year by year (9^{th} (fresh), 10^{th} (soph), 11^{th} (junior), 12^{th} (senior)) in high school. They know that they lose students each year — there is, one might say, "leakage" out of the 9^{th} grade class — facing various problems, some of the kids simply drop out of school. And the same for 10, 11, and 12.

What are the stocks?

What are the flows?

What are the "clouds" in this model?

Draw the stock and flow diagram showing stocks, flows, valves, clouds.

Next, let's assume that there are aggregate drop out rates (DR) for each year. That is, there is a DR_{1} that is the drop out rate during year 1, etc. REDRAW diagram with the DRs shown as "information" or "settings" with arrows connecting them to the valves they influence.

Your principal tells you that they have been collecting data on the sorts of challenging issues kids face — getting arrested (LAW), getting pregnant (PREG), and having academic troubles (ACAD) — and they have ideas on how prevalent each problem is at different years and how it affects drop out rates. In other words, they have numbers such as LAW_{1} which describes what percentage of freshman get in trouble with the law each year, or ACAD_{3} which is the percentage of juniors who have academic difficulties. Add LAW_{1}, LAW_{2}, LAW_{3}, LAW_{4}, PREG_{1}, etc. to the diagram with arrows connecting them to DR_{1}, DR_{22}, etc.

If DR_{1} is the fraction of 9^{th} graders, (P9) who drop out (for all reasons), write an equation for P10, the number of 10^{th} graders.

SIMPLE PROBABILITY

# MC01: Demonstrate understanding of random selection from different shape distributions (uniform, normal, Poisson)

Suppose we have two groups of people. In one group, height is uniformly distributed. In the other group it is normally distributed.

# MC02: Apply basic concepts of probability (events and outcomes, and, or, conditional)

Consider the rolling of an ordinary, fair, six-sided die. How many outcomes are there?

What is the probability of the event "Do not roll a five"?

What is the probability of three successive rolls without getting a five?

Would you accept a bet that said "if you roll three times and don't get a five I'll give you $10, but if you roll any fives you give me $10"? Explain your reasoning

# MC03: Set up difference equation version of a Monte Carlo simulation

We are given the following probability distribution

20% of the time, no clients arrive in a given hour

30% of the time , 1 client arrives per hour

30% of the time, 2 clients arrive per hour

20% of the time 3 clients arrive per hour

A | B | C | |

1 | TIME | RANDOM NUMBER | NUMBER OF ARRIVALS |

2 | 1 | =rand() | = |

3 | 2 | =rand() |

# MC04: Use rand() and related functions and data table function in Excel to generate Monte Carlo simulation

In Excel, the rand() function delivers a random number between 0 and 1. We would like to build a simulation of a die roll — we want the outcomes to be 1,2,3,4,5,6. What would your approach in Excel be?