What Are the Chances?

If making decisions is part of your job, you know that some decisions are far from straightforward. Sometimes we make decisions based on partial information or guesses. For example, when we must choose between A and B, we make estimates of the probabilities of A's success and B's success. Then we pick the one we think best. Sometimes we do this without realizing that this is the process we're using. That's where it gets interesting, because most of us, as humans, are really bad at making decisions that involve probabilities.

This post People make a common mistake
when forming estimates of
the probability of multiple
events occurringprovides an example of a simple decision problem involving probabilities. It's a problem that causes many of us to take a wrong turn. I begin by posing the problem in semi-realistic terms fairly close to something that actually could come up at work. After posing the problem, I take a short excursion into a little refresher of some basic probability methods. Then I'll return to the semi-realistic problem to apply those methods. Should be fun.

A semi-realistic problem

Suppose we're making a risk plan. The risk we're planning for is the risk that members of our team will be stricken with COVID during the next month. For this scenario, an important fact is that our team members are all classified as essential, and we all work in the same suite of offices, using the same break room. That's why we're concerned about COVID.

Our plan is to find a way to meet our deadlines as best we can, even if some of our team fall ill. So we do a bit of cross training and we prepare for several scenarios. Here's our plan:

• If A falls ill, B and C will take over A's responsibilities
• If B falls ill, C and A will take over B's responsibilities
• If C falls ill, A and B will take over C's responsibilities

We estimate that the probability of anybody falling ill during the next month is 10%. So the question is: How well will our plan work?

One thing is clear: if all of A, B, and C fall ill, our plan won't work at all. So our task is to work out the probability that all three of A, B, and C will fall ill.

A little refresher about probabilities

Our semi-realistic problem has three events: A falls ill, B falls ill, and C falls ill. That's a little complicated, so let's take a look at a two-event problem. One event is this:

Event X: When the Sun sets tonight, at the moment of sunset the sun will be obscured by clouds.

The probability: 1/5

And the second event is this:

Event Y: Right after sunset, I roll a die, and it lands so that either one or two spots are showing.

The probability: 1/3

So the question is this: what is the probability that both Events X and Y occur? Call that XY.

Solution: P(X) is 1/5. P(Y) is 1/3. So P(XY) = P(X) times P(Y) = 1/15.

Explanation: X occurs one-fifth of the time. For the other times, it doesn't matter what I get when I roll the die. Only when X occurs does the die matter. So right from the beginning, we're down to P(X)=1/5 of the time as an upper limit for P(XY). But the probability that Y occurs is only 1/3. So P(XY) = 1/3 of 1/5 of the time. That is P(XY) = P(X)*P(Y) = 1/15.

So here's what we learn from this: If Event X and Event Y are independent of each other, P(XY) = P(X) * P(Y). This isn't a proof. It's a plausibility argument. I'll leave the proof to the mathematicians.

It's also plausible to suggest that the number of independent events doesn't matter. If I add a third event Z:

Event Z: I flip a coin and it lands heads side up

The probability: 1/2

Then the probability of XYZ = P(XYZ) = P(X)*P(Y)*P(Z) = 1/15 * 1/2 = 1/30.

The generalization of what we've learned is this:

The probability that every one of a set of independent events occurs is the product of the probabilities of the members of the set of events.

Now we're ready to return to our semi-realistic problem.

Solving our semi-realistic problem

To compute the probability that our risk plan will fail totally, many would argue as follows. If P(A), P(B), and P(C) are respectively the probabilities for A, B, and C falling ill, then the probability of all three falling ill would be P(A,B,C) = P(A)*P(B)*P(C) = 0.1*0.1*0.1 = 0.001. So by this calculation, our plan will be workable 999 times out of 1000.

Unfortunately, in the real world, that would be wrong. By a lot.

What's wrong with our reasoning is this. To find the probabilities of all of A, B, and C falling ill, we multiplied their respective probabilities together. The theorem we're applying (actually misapplying) is, as we saw above:

The probability that every one of a set of independent events occurs is the product of the probabilities of the members of the set.

But in this case, the probabilities of the three events aren't independent.

That is, the probability that B falls ill, given that A is ill, is no longer just 10%, because all three work in the same office. One can infect the others. We might not know what the probability actually is, but it's more than 10%. The same is true for C. The probability that C falls ill, given that A is ill, is greater than 10%. Computing the actual result is difficult, but we can tell right away that the probability of all three falling ill is much larger than 1/1000.

Last words

If you surf the Web much, you'll find numerous examples of misapplication of our probability rule for independent events. The rule works well for independent events. But to apply it to multiple events needs justification. Verify that the events are independent before using the method.  Top    Next Issue

Projects never go quite as planned. We expect that, but we don't expect disaster. How can we get better at spotting disaster when there's still time to prevent it? How to Spot a Troubled Project Before the Trouble Starts is filled with tips for executives, senior managers, managers of project managers, and sponsors of projects in project-oriented organizations. It helps readers learn the subtle cues that indicate that a project is at risk for wreckage in time to do something about it. It's an ebook, but it's about 15% larger than "Who Moved My Cheese?" Just . Order Now! .

Your comments are welcome

About Point Lookout

Thank you for reading this article. I hope you enjoyed it and found it useful, and that you'll consider recommending it to a friend.

This article in its entirety was written by a human being. No machine intelligence was involved in any way.

Point Lookout is a free weekly email newsletter. Browse the archive of past issues. Subscribe for free.

Support Point Lookout by joining the Friends of Point Lookout, as an individual or as an organization.

Do you face a complex interpersonal situation? Send it in, anonymously if you like, and I'll give you my two cents.

Related articles

More articles on Problem Solving and Creativity:

Comfortable Ignorance

When we suddenly realize that what we've believed is wrong, or that what we've been doing won't work, our fear and discomfort can cause us to persevere in our illusions. If we can get better at accepting reality and dealing with it, we can make faster progress toward real achievement.

Clueless on the Concept

When a team member seems not to understand something basic and important, setting him or her straight risks embarrassment and humiliation. It's even worse when the person attempting the "straightening" is wrong, too. How can we deal with people we believe are clueless on the concept?

Group Problem-Solving Tangles

When teams solve problems together, discussions of proposed solutions usually focus on combinations of what the solution will do, how much it will cost, how long it will take, and much more. Disentangling these threads can make discussions much more effective.

Cost Concerns: Comparisons

When we assess the costs of different options for solving a problem, we must take care not to commit a variety of errors in approach. These errors can lead to flawed decisions. One activity at risk for error is comparing the costs of two options.

On Reporting Noncompliance

Regulating compliance with process design in organizations requires monitoring process usage. Typically, process monitors depend on reports from process participants. In blame-oriented cultures, fear of retribution can limit what these reports contain.

See also Problem Solving and Creativity and Critical Thinking at Work for more related articles.

Forthcoming issues of Point Lookout

Coming October 4: Self-Importance and Conversational Narcissism at Work: I

Conversational narcissism is a set of behaviors that participants use to focus the exchange on their own self-interest rather than the shared objective. This post emphasizes the role of these behaviors in advancing a narcissist's sense of self-importance. Available here and by RSS on October 4.

And on October 11: Self-Importance and Conversational Narcissism at Work: II

Self-importance is one of four major themes of conversational narcissism. Knowing how to recognize the patterns of conversational narcissism is a fundamental skill needed for controlling it. Here are eight examples that emphasize self-importance. Available here and by RSS on October 11.

Coaching services

I offer email and telephone coaching at both corporate and individual rates. Contact Rick for details at rbrenogMhuqCxAnbfLvzbner@ChacigAthhhYwzZDgxshoCanyon.com or (650) 787-6475, or toll-free in the continental US at (866) 378-5470.

Get the ebook!

Past issues of Point Lookout are available in six ebooks:

• Get 2001-2 in Geese Don't Land on Twigs (PDF, )
• Get 2003-4 in Why Dogs Wag (PDF, )
• Get 2005-6 in Loopy Things We Do (PDF, )
• Get 2007-8 in Things We Believe That Maybe Aren't So True (PDF, )
• Get 2009-10 in The Questions Not Asked (PDF, )
• Get all of the first twelve years (2001-2012) in The Collected Issues of Point Lookout (PDF, )

Are you a writer, editor or publisher on deadline? Are you looking for an article that will get people talking and get compliments flying your way? You can have 500-1000 words in your inbox in one hour. License any article from this Web site. More info