APPLICATION OF PROBABILITY TO ASSESS RISK IN MANAGEMENT DECISIONS, RISK ANALYSIS

Jim Kennedy

CONCEPTS

Quantitative Risk Analysis is a tool used to aid in management decisions when uncertainty has to be considered.

A mathematical equation is a model composed of one to an infinite number of variables, and uncertainties.

VARIABLE OR UNCERTAINTY

Variables are controllable Weight/volume of a chemical used in a

reaction Amount of antibiotic administered Cost of a diagnostic test Sensitivity and specificity of a diagnostic

test ? ?

UNCERTAINTIES

Uncertainties are uncontrolled but predictable. Prevalence Immune response When will a shear pin break Who will win the final four

PREDICTION=PROBABILITY=UNCERTAINTY

Uncertainty is the level of ignorance. Uncertainty is lessened by knowledge

Literature reviews Expert opinions Further data gathering and analysis

Until an uncertain event becomes 100% controlled it is a function of probability.

Uncertainties, like variables, may be discrete or continuous.

Whether an unborn calf is male, female, or a hermaphrodite would be a discrete uncertainty.

The weight of a calf at birth is a continuous uncertainty, it could weigh 75.0 lbs 75.1, 75.15…etc. an infinite number of possibilities.

GRAPHICAL REPRESENTATIONS OF UNCERTAINTIES

Discrete uncertainties and their probabilities are more easily understood if represented visually by a bar graph.

Continuous uncertainties and their probabilities are more correctly represented by a line.

A graphical representation of an uncertainty and its probability may be cumulative or may be a single point.

SIMULATION MODELING

Using probability to make a management decision. Putting together a series of variables and

uncertainties into a mathematical formula produces a model.

Values to each variable and uncertainty can be given and the outcome of the mathematical formula be determined, a deterministic model.

An alternate method is to assign probabilities to all or some of the uncertainties and allow the probability distribution to determine a mean value , upper and lower limits, and a standard deviation for the uncertainty.

STOCHASTIC MODELING

Accounts for an uncertainty occurring dependent on the probability of that uncertainty. We are uncertain of the prevalence of a disease

within a herd, but we can make a guess and assign a probability to that guess.

Your best guess is that 10 of 100 animals are infected, but you know that it is possible that none are infected or that all are infected, you are uncertain. If the decision to be made is metaphylaxis or not,

a stochastic model might allow the best decision be made.

MAKING THE MODEL

Assuming your decision rests on whether metaphylaxis would be more cost effective than to pull and treat you would consider factors such as: the cost of metaphylaxis the cost to treat the ability of the pen rider to pull sick animals the number of head requiring treatment how many animals will require treatment despite

metaphylaxis how many animals die although treated etc.

MODELS GET COMPLICATED QUICKLY

Most of the factors for consideration from the previous slide are uncertainties A model with too many uncertainties may produce

invalid results, you may end with odds of making the correct decision based on the model of 50:50, you didn’t need a model just a coin.

To avoid the dilemma you make assumptions such as the pen riders are the best or the treatment never fails, but assumptions decrease the validity of the model.

OTHER ASSUMPTIONS

Besides the assumptions of the model you also make assumptions about the probability distribution of the uncertainties.

The more precisely the probability distribution of an uncertainty can be defined the more precisely the model will depict reality.

SOME PROBABILITY DISTRIBUTIONS OF INTEREST

Pert(3.0000, 8.0000, 10.000)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

2.0

3.3

4.6

5.9

7.2

8.5

9.8

11.1

5.0% 5.0%90.0%5.226 9.370

Pert Distribution

Normal(7.0000, 1.5000)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

3 4 5 6 7 8 9 10 11

< >5.0% 5.0%90.0%

4.533 9.467

Normal Distribution

HyperGeo(20, 12, 100)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-2 0 2 4 6 8 10

12

14

5.0%90.0%

0.00 5.00

Hypergeometric Distribution

Uniform(0.0000, 50.000)

V

alu

es

x 10^-

2

0.0

0.5

1.0

1.5

2.0

2.5

-10 0 10

20

30

40

50

60

90.0%2.50 47.50

Uniform Distribution

SIMULATION MODELING

Simply stated a random set of values are placed into a mathematical formula and the results recorded.

A list of values could be placed in a spreadsheet and selected at random this might not reflect the probability of the value actually occurring.

Different methods of random selection of the values such as Monte Carlo or Latin Hypercube sampling exist.

RANDOM SELECTION FOR SIMULATION MODELS

Monte Carlo and Latin Hypercube simulation interpose the probability distribution of the event on the selection of the random value.

The resulting difference between the two methods in most cases is minor, Latin Hypercube is faster (requires less cpu time) than the Monte Carlo method.

AVAILABLE AT A PRICE

Software programs to do simulation modeling are available, such as @Risk and Crystal Ball.

These programs are pricey and offer some challenge in applying.

Programs/software of this type are used by industries and governmental agencies in decision making. I would suspect that one of these programs may have been used to reach a decision on the Wall Street bailout, or if not it should have been.

PREGNANCY TESTING 500# FEEDLOT HEIFERS

Question: Would it be more cost effective to pregnancy test 5000 500# heifers or do nothing? Cost to pregnancy test Lost revenue for pregnant heifer Prevalence of pregnant heifers in group

Which are variables and which are uncertainties?