The Chance of Rain
Most people do not understand what the chance of rain means. Some believe it means it will rain "30% of the day," others "in 30% of the area," while others “when the weather conditions are like today, in 3 out of 10 cases there will be (at least a trace of) rain.”
In my webinars and workshops over the past decade, the usual audience split is 30 percent for each alternative (roughly 10 percent respond incorrectly that the chance of rain has something to do with the amount of rain that will fall).
This means that two-thirds of any audience does not understand probability.
Foundations - Probability Defined
The short definition in Merriam-Webster's Dictionary is "the chance that a given event will occur.”
The long definition is “the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes.”
The term ratio (or percentage) can add to the confusion. Ratios can be back-sighted for events that have occurred (such as results of experiments) or front-sighted for events that may occur in the future (such as results from Monte Carlo analysis). Many audiences are confused by the percentage of a past event (whose probability is 1 since it has occurred) and a percentage [prediction of what may occur in an uncertain future (something less than 1).
Baseline - Prediction is a Fool’s Game
Every event that refers to future occurrences is uncertain. What we refer to as probability reflects our current knowledge. Probability is simply one valid method to express our degree of certainty (or uncertainty) in quantitative terms. Only clairvoyants and fortune tellers can predict the future with complete certainty.
Rational thought as defined by objective numerical analysis is a modern one. Many technical professionals were trained in statistical hypothesis testing and consequently (and usually unconsciously) have a “baked-in” statistical Frequentist tendency. It is simply flawed thinking that a conclusion drawn from the past will objectively predict the future.
Probability is based on our current knowledge of future uncertain events, which always makes probability subjective.
Helpful Tip – Consider Compound Probability
Compound probability is the likelihood of more than one event occurring at the same time or the probability of event A and event B occurring together. There are two underlying conditions:
Events A and B must happen at the same time (for example, throwing two dice simultaneously).
Events A and B must be independent of each other (event A outcome does not influence event B outcome).
If these two conditions are true, the probability of event A can be multiplied by event B.
In other words, if it rains on 50% of the days like today and I usually have a 50% chance of catching more than 10 fish when I fish in the morning, then there is a 25% chance that I will catch fish this morning (0.5 times 0.5).
This probability assumes that I have good historical data (maybe “yes” on the rain forecast but likely “no” on tracking the fish count). And it assumes that the weather and catching fish are independent.
The tip is to list the underlying events and determine whether they are independent of each other. Then take a look at your historical data of each past event. Communicate this information before stating the probability of your prediction.
Helpful Tip – Consider Conditional Probability
Conditional probability is the likelihood of one event occurring, given that another event has already occurred. In other words, the idea of independence is now broken.
In our example, catching more than 10 fish in the morning depends on whether it has rained.
The math follows Bayes theorem, which we will not derive here, but it comes down to knowing the probability that it will rain and the probability of how often you catch 10 fish when it rains. We have to have better data and make a major assumption of whether it has rained in advance of knowing it rained.
The tip is to note all of the events that must occur before a predicted event is to occur. Then decide whether some of the underlying events must have happened to produce the predicted event or if all the underlying events occurred simultaneously to produce the predicted event.
Communicate all of the underlying events and the certainty (the data of past events) you have before stating the probability of your prediction. Then indicate whether the underlying events were independent of one another.
Helpful Tip - Comparative Comparisons
Organizations usually avoid after-event reviews. These reviews include predictions performed by modelers, data scientists, mathematicians, statisticians, engineers, and other enlightened staff. One of the issues is that it takes time and seasoned resources to do an autopsy. The other part is that most of us, including senior management, do not really want to know how bad our probability predictions were.
To communicate probability effectively, you must be able to communicate how good or bad you (or your industry) have been doing as a whole. If you lack that data, communicate historical values that forecast models are 60 to 80 percent accurate.
That’s right, many predictions of an uncertain are not much better than the flip of a coin. In some cases, like selling music, a twenty percent variance is not bad; however, it is wholly unacceptable in healthcare, safety, and finance.
The audience needs to understand how good or bad we are at computing or assigning “the chance that a given event will occur.”
Helpful Tip – Sensitivity Analysis
A sensitivity analysis conveys the relationship between inputs and outputs. Two common ways to visualize the sensitivity analysis results are with a cone diagram or a tornado diagram.
A cone diagram is used to depict the results of a probabilistic forecast. The cone diagram resembles a funnel with its narrow end at the present time and its wide end at the end of the forecast period. Normally, a different shade of color is shown to distinguish the central 50th or 90th percentile from the extreme outer ranges. The cone diagram is easier for most decision makers to understand.
Tornado diagrams are modified versions of a bar chart. They are a classic tool used to communicate the results of a sensitivity analysis, which can be performed either deterministically or probabilistically. The tornado diagram receives its name from the visual image created from wider bars associated with input variables that have more impact on the output being located at the top, while the narrow bars associated with input variables with less impact on the output are shown at the bottom.
Explaining the results of a single sensitivity analysis takes time. The best manner to communicate effectively is to show two or more cone diagrams or tornado diagrams on the same visual. Senior management understands that wider ranges are bad. The presentation will appropriately and quickly migrate to questions about which inputs are causing the most variable scenarios, whether or not the decision makers completely understand the visuals.
I’m Going Fishing
Well, it’s time to go fishing now. The probability seems high that I will catch fish on a day like today – it is sunny, the wind is from the southeast, the windspeed is between 5 and 10 miles per hour, and the ocean is calm. I guess I should also factor in that the tide is rising. I do not have fresh bait or the best type of bait that the fish feeding like, and it looks like someone else is sitting on my favorite hole (there is a good bottom structure for fish).
There are indeed many factors that go into defining what a day like today is. Or how all of those factors may change in an hour. Perhaps the more I know about the past, the more accurate my prediction may be.
The more I know about probability, the more I understand that there is not much certainty in assigning a probability to an uncertain future.
Two-thirds of your audience will not understand what the chance of rain means. In the limited time you have, it isn't easy to attempt to create an understanding of probability while at the same time communicating a future prediction. So don’t.
Instead, communicate the underlying events, the independence (or dependence) of each with the other, and the amount of historical data you have about each underlying event. You will be telling your audience about your certainties. Or even better, you will be indicating your uncertainties.
Also, give the autopsies of how good past predictions of the same thing have been. If you do not have that data for you, your organization, or your industry, then that is telling your audience something too.
Finally, show them a sensitivity analysis of the input variables. Comparing tornado diagrams (bar charts) and cone diagrams (avoid box and whiskers) of different scenarios will help provide an understanding of which input variables are most critical.
Remember the obvious, too. Most senior managers understand that predicting the probability of future uncertain events is a fool’s game.
When communicating probability, avoid trying to buck instincts and convince your audiences of the certainty of the prediction. Communicate the uncertainties early and often.
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