In times of decision, many of us rely on instinct and ‘gut’ reaction rather than logic. As humans, we actually do a pretty terrible job at understanding and using risk-based approaches in our lives. For example, many people are anxious about flying on an airplane when in reality they should be more scared about the car trip to the airport. We allow inputs from our environment (such as the news) and preconceived notions to dictate how we view and act in relation to risk.
The concept of threat intelligence aims to identify areas of security risk by allowing organizations to focus on true vulnerabilities, exposures, and the likelihood of risk realization. This provides actionable intelligence based on inputs and applied values, and is generally accepted as accurate enough to avoid or minimize risk exposure. The intelligence is of little use if we improperly value input factors, choose to ignore the results, or willfully allow our preconceived notions to bend or distort logic.
Here is a quick example of how we twist true logic into what we think it should be. This is a simple mathematical proof highlighting how we sometimes use preconceived notions to guide us into making a wrong decision.
The Game Show
This exercise in logic is based on a game show, in which the contestant is required to pick one out of three identical closed doors. Behind two doors there is a goat, but behind the third is a car. The contestant must pick only one door. If they pick correctly they win a new car. If they pick incorrectly, they get a goat (I think…I never really followed up on that). I assume a goat-choosing contender gets to feed it until pawning the beast off to a kid’s petting zoo.
The problem has this added twist: after the contestant has made a selection, the host opens one of the two remaining doors (the ones that the contestant did not pick), and behind that door is a goat. The host then asks the contestant if he wants to switch his choice or remain with the initial choice. The contestant may change their original choice or stick with what they have. Then all three doors are opened, and if the contestant made the correct choice, he or she gets to drive home a new car (hopefully in style and comfort). Or figure out got transportation requirements with the TSA.
There are many variations on this, all of which may or may not affect the outcome. I have described the most common variant in our example game, and it assumes the following:
The contestant wants to win the car and therefore will act to maximize the chances of winning.
The host always opens one of the doors that the contestant has not picked, and shows a goat behind it.
The contestant is then given the chance to select either of the two remaining closed doors.
This problem created a controversy because of a counter-intuitive subtlety that was introduced into the initial statistical reasoning. Initially, the contestant had a 1 in 3 chance (33%) of picking the correct door, since all 3 doors are identical and there is no way to tell which has the car behind it. Therefore, the chance to win is 33% and your chance to lose is 66%.
Moving on…we assume the contestant picks the middle door as their choice.
The host will always reveal a goat behind door 1 or 3.
Now the contestant has the option of sticking with their choice of door number 2 or changing their choice to the remaining closed door.
A contestant may conclude they should stick with their instinct. We created this test as a Web game on DrChaos.com in 2014 and found the majority of players kept their original choice. When we asked them why they reasoned they had a 50/50 of winning the car (there was no real car to win…sorry we don’t have sponsors. Yet).
Is That True?
First of all, let’s examine the initial choice. The contestant started out with a 33% chance of picking the correct door.
When the host opens one of the doors that the contestant did not select and shows a goat behind it, many people assume that nothing has changed so far as the problem goes. Each door is still associated with a 33% chance of having the car, therefore the two remaining doors (the one that the contestant initially chose and the other which the host did not open) still have a 33% chance of having the car.
Generally, people reason that the opened door has changed the probabilities of the two remaining doors. Since the car HAS to be behind one of the two remaining doors, they divide the probability between them, assigning 100/2 = 50% probability to each of the remaining doors. What they have actually done is to take the open door and distribute its probability equally between the two remaining doors.
Before the host opened the door, it had a 33.3% probability of having the car, the same as any other door. After the host opened the door and showed a goat behind it, the probability went from 33.3% to 0% for that door. People then tend to distribute this probability equally between the two remaining closed doors, adding 16.6% probability (33.3% divided by 2) to each of the remaining doors, making the probability 33.3+16.6 = 50% for each of the remaining doors.
Since they have assigned an equal probability to the remaining two closed doors, there is no clear reason to accept the host’s choice to switch. Both doors now seem to have a 50% probability of winning the car, and the contestant’s odds are not improved by switching.
However, they were wrong!
Why?
The contestants correctly recognized that the initial probability of each door having a car is 33% per door. They further correctly identify that when the host opens one door and shows a goat behind it, the odds on the remaining doors have changed. Information has been added to this probability ecosphere, and this information must be taken into account when calculating the new probabilities. The mistake is in how the new probabilities are calculated.
If you simply divide the open door’s initial probability between the two remaining doors (making 50% each for the remaining doors as explained above), you have acted upon an incorrect assumption! You are inferring the host opens one of the two doors at random. If this were true, then the host would pick the door with the car 50% of the time. But this simply does not happen.
According to the rules, the host always opens a door and shows a goat behind it. In other words, the host has knowledge of which door hides the car (and deliberately avoids it), and this knowledge has also entered this probability ecosphere. Let’s take a quick glance at the host’s constraints:
He cannot open the door the contestant selected, because then he would not be able to offer the choice to switch.
He cannot open the door with the car, or the game is over, and again he can’t offer the contestant the choice to switch.
This means the host has to pick one of the doors the contestant has not selected and which does not have a car behind it. The two doors that the contestant did not pick had a combined probability of 66% of having the car, as can be seen on the first diagram. When one of them is opened and a goat is seen behind it, that door’s car probability goes to 0%. The remaining door assumes the entire 66% probability which the two doors had in sum before one was opened.
Picking a door at the start means you have a 66.66% chance of being wrong. That 66.66% probability is split between the two doors you didn’t choose. The host opens one of these doors but is constrained to open only the one which has a goat. Therefore, your 66.66% probability of being wrong now moves to the single remaining door which you didn’t pick and the host didn’t pick.
You can now change your choice and turn that 66.6% chance of being wrong into a 66.6% chance of being right.
So after the door is opened to show a goat, the contestant has a 33% chance to be right when keeping the original choice, and a 66% chance of being wrong. Therefore, you can double your chances of being right if you accept the switch.
It still seems counter-intuitive to the original odds but yes, the results of the test game we ran on DrChaos.com were consistent with the statistical reasoning presented here.
In Conclusion
If you follow the odds as stated above and end up with the goat, please treat it nicely. The odds were not favorable for the goat to gain a new friend. The universe spoke. Go with it.
Special shout to the Fort Collins Rescue Mission, and to Chris and Belle.
