Amazing Applications of Probability and Statistics

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How to Design Small Decision Making Groups


Teamwork and group decision making are hot buzz words. Giant corporations spend millions annually teaching employees how to work in groups and hold effective meetings. Yet, anyone who has attended a decision making meeting probably feels that a camel  is indeed a horse designed by a committee. When decision making teams work, they work very well and when they don't they consume inordinate amounts of time only to yield weak results. Simple applications of probability and statistics can shed light on how to stack the odds in favor of success.

This discussion will focus on decision making groups working with partial information where there is a definite difference between selecting a good and bad alternative. For example, this could be a decision about how many toys to produce for the Christmas market. There is only partial information since actual customer behavior cannot be known until well after the decision is implemented. Making the wrong number of toys can result in a substantial loss of profit.

Such groups have two key problems. The first is management of communication. The second is decision making accuracy. Generally having more people in a group increases the likelihood that someone will propose the correct decision. However, more people means more opinions and ideas that have to be communicated and discussed. This makes management of the communication process more difficult and can end up reducing group effectiveness. The best ideas may never even be heard.


Management of Communication

The difficulty of managing communication is roughly proportional to the number of possible social interactions. With two people there is nothing to manage since there is only one possible social interaction. With three people there are three possible two-person interactions and one three way interaction for a total of four. With a four person group there are six possible two way, four possible three-way, and one possible four way interactions for a total of eleven.

Rules for Optimizing Small Groups

  1. Group members must have the basic knowledge and ability to identify  and evaluate alternatives. Not everyone needs to be an expert but everyone must be able to ask intelligent questions.
  2. Each member must be granted equal status or standing within the group. A single member may be the only one to have the correct solution. Equal status improves the chances that this solution will be expressed.
  3. Each member must be allowed to fully explain and argue for his or her proposed solutions. This is the only way a single individual with the correct solution can convince the group.
  4. Criticisms of options should be made only after all options have been listed. This helps insure that all options are identified.
  5. For major decisions, discussion should proceed until there is unanimous agreement that a workable solution has been reached. Requiring a unanimous agreement forces a greater amount of discussion and gives a single individual with the correct solution the opportunity to convince others. Unanimous agreement can involve compromise. It means that all the members have agreed to a workable solution although it may not be their personal favorite.
  6. For minor decisions, the majority rules. This speeds up group activities without a major loss of effectiveness.
  7. Groups should have an odd number of members. This prevents ties and improves the odds of making a correct decision when using majority rules.
  8. The optimum group size is about 5 members. Groups should be expanded beyond this size only when there is good reason. For instance, effective implementation may require buy-in by more than 5 major stakeholders.
The total number of possible social interactions for any sized group is simply the sum of all possible combinations taken two at a time or higher. These are calculated for 2-person interactions using the following Formula:


x!(n - x)!


As can be seen by figure 1, the number of possible social interactions begins to explode in groups with more than 5 people. Large groups require skillful leaders and formal structure in order to function effectively. Formal structure, such as parliamentary procedure, works by deliberately stifling many of the possible social interactions. Unfortunately, this can also stifle creativity and insures that decisions will be dominated by the most politically skillful individuals even when they don't have the best ideas.

Best Possible Decision Making Accuracy

Probability can also be used to evaluate decision making accuracy. This can get complicated so we'll make some simplifying assumptions as follows:

  1. There is partial information available on all possible alternatives. Obviously, if there's no information available then decision accuracy is random. If complete information is available then the decision accuracy should depend only on having enough time and expertise to evaluate the information.
  2. All alternatives can be clearly categorized as good or bad. Those categorized as good are clearly superior to those categorized as bad. Selecting a good alternative is a good decision by definition. There can be no ties. A decision is either good or bad.
  3. The probability of successfully identifying a superior alternative is not influenced by the number of alternatives. If bad alternatives greatly outnumber good ones then it does not make the good ones harder to identify. In real life many bad alternatives can be quickly eliminated.
  4. Each group member has the same probability of successfully identifying a good alternative.  A decision making group should contain qualified people who have the ability to discuss and analyze the alternatives. When only partial information is available there typically are no experts with the special ability to consistently guess the right answer. If such an expert existed then there would be no accuracy advantage in using a group.

Divorce statistics for first marriages provide information on the average probability of picking a good alternative in an important decision using partial information. Failure to pick a good alternative can be defined as divorce. Overall, after 20 years 50% of all first marriages have ended in divorce. 

Based on the divorce statistic, it looks like 50% would be a good number for the average decision making accuracy of typical individuals. However, closer examination indicates that 50% is a low estimate. Women who marry when they are less than 18 years old have a 67% chance of ending their first marriage within 20 years. On average, members of this group have not fully developed their ability to identify  and evaluate alternatives. Their higher divorce rate skews the average results for all marriages to the low side. 

Women between 20 to 24 years of age, on the other hand have a 41% chance of divorce within 20 years. This rate appears to remain approximately stable for women who marry at 25 years of age or older (see Figure 1A). Based on this information 60% accuracy looks like a better rule of thumb for adult decision makers who are using partial information.

Adoption of the 60% average accuracy rule of thumb is not based on a mathematical proof but is reasonable. In many cases a 50% average accuracy could be achieved by coin tossing. It's unlikely for human decision making to be worse on average than 50% accuracy unless fraud or deception is involved. 60% accuracy seems more reasonable without being overly optimistic about humankind's abilities.

Assuming an average decision making accuracy of 60% allows us to calculate the probability that no one in a group will identify a good alternative. Obviously if this were true then the group could not possibly make a good decision. The probability of no one in the group identifying a good alternative is found as follows:

P0 = (0.4)n
Where n = the number of people in the group.

The best possible decision making accuracy of the group would be Pg = 1 - P0 .  

Figure 2 shows a graph of the best possible decision making accuracy vs group size. To achieve this level requires that the decision be made unanimously following an extended discussion. The Rules for Optimizing Small Groups shown at the top of the page must be followed to achieve this performance. These rules are designed to insure that all options are identified and that they are fully discussed. If the group is composed of qualified individuals with the basic knowledge to make the decision and they behave in a rational manner, a single enlightened group member should be able to persuade the group to select the right alternative. On this basis the only way the group would fail is if no one in the group selects a good alternative.


Majority Rules Decision Accuracy

There is a tendency in groups to make decisions by voting, in which the majority rules. While this speeds the decision making process it reduces the accuracy and should be used only for minor decisions.

To study this situation we resort to finding areas under the binomial distribution using the following relationship:

S ( n! ) (qn-x)px
x!(n - x)!
n = number of group members
m = the minimum size for a majority
p = the probability of being right or .6
q = the probability of being wrong or .4 
note: q = (1-p )

Figure 3 displays the results of majority rules decision making for various sized groups. Note the odd looking saw tooth appearance which gives even numbered groups a lower probability of making the correct decision. The explanation is that these groups can have a tie. For example, a group of four will only have a majority 75% of the time. The other 25% of the time will be a tie vote which obviously does not result in a correct decision. Odd numbered groups will have a majority on every vote. Yes, even number groups can often resolve tie votes but it  costs more time and effort to do it.



Figure 4. shows that a group size of five is optimum. Five takes advantage of the desirability of odd numbers for majority rules decisions. For the unanimous decision making style a group of five will have a 99% accuracy assuming 60% individual accuracies and that a single person with the right answer can convince the others. Even with only 50% individual accuracies the group accuracy will average 96.9%. Adding additional members will not greatly improve accuracy. However, additional members will significantly increase group management problems since the number of possible social interactions increases rapidly.

The data presented does not mean that every group has to contain exactly five members. There are other factors to consider such as the implementation of decisions. This often requires buy-in by the stakeholders. Placing stakeholders in the group can speed implementation. However, important decision making groups should not be expanded unless there is an overriding reason to do so.



1. Matthew D. Bramlett, Ph.D., and William D. Mosher, Ph.D., Division of Vital Statistics, "First Marriage Dissolution, Divorce, and Remarriage: United States", Advance Data Number 323 +May 31, 2001Department of Health and Human Services

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