Puzzles on this page:
The Game Show, the Donkeys and the Car
Revolver
Cells of Drug Probability
The Game Show, the Donkeys and the Car
This is a fascinating conundrum that has puzzled people through the ages.
However, behind each of the other two remaining doors sits (or rather stands) a tethered donkey. This problem does assume you would wish to win the car and not one of the donkeys!
The Quizmaster now asks you to pick a door to which you think (or rather guess) may be the door that conceals the star prize luxury car. For the rest of this game we will conduct by example as any of the 3 doors could be picked. In this example, you decide to pick door A and tell the Quizmaster so. The Quizmaster, who knows what is behind each door now opens door C, revealing a tethered donkey. The Quizmaster now says you have another and final chance and asks whether you want to stay with your original choice of door A, or change your mind and now go with door B?
What do you do? Stick with door A or change your choice to door B!
Have a think and decide what you would do, before reading further.
Revolver
Just like The Game Show, the Donkeys and the Car, this Revolver exercise defies initial logic by reducing a probability when logic says the probability should increase! In the case of The Game Show, the Donkeys and the Car, the chances of winning double by changing your mind, when nothing else has changed – or so it seems! On to the Revolver.
Take a 6-chamber revolver and put in 2 bullets at random spinning the chamber barrel each time so having no knowledge as to where the bullets are in relation to each other or the firing pin. What are the odds of the gun firing if the trigger is pulled?
Quite a simple probability problem this as there are 6 chambers and 2 bullets making the odds or probability of 2 chances in 6 or 1 in 3 or 33.3% of the likelihood of the gun firing. That means that there is a 66.6% chance of the gun not firing. So, there is a greater chance of the gun not firing than firing. Agreed?
So, let’s assume the gun does not fire. Now, without spinning the chamber, what are the chances now of the gun firing if the trigger is pulled? There are now 5 unused chambers left and still the 2 bullets. So, the chances of the gun now firing are 2 in 5 or a 40% chance of the gun now firing. As would be expected the chances of the gun firing have increased. However, let’s assume the gun does not fire on the second trigger pull and the chances of the gun firing on the third trigger pull are now 2 in 4 or 50%. Again, let’s assume the gun does not fire on the third trigger pull and the chances of the gun firing on the fourth trigger pull are now 2 in 3 or 66.6%. If the gun still doesn’t fire on the fourth trigger pull, the next trigger pull will definitely fire with 100% probability as there are only 2 chambers containing the 2 bullets left, which happen to be in adjacent chambers. So, don’t trigger the gun a fourth time, as we know it will fire!
Leaving the bullets where they are, in their adjacent chambers, now spin the chamber barrel quickly and many times so that we do not know where the bullets are in relation to the firing pin.
We are now beginning a new trigger pulling sequence without spinning the chamber barrel, but letting it rotate one by one with each trigger pull.
What are the chances of the gun firing on the first trigger pull? As before 33.3%. Agreed?
Again, assume the gun does not fire on the first trigger pull. The question is this;
Are the chances of the gun firing on the second trigger pull higher or lower than 33.3%?
What do you say?
Most people say that the chances of the gun firing on the second trigger pull are higher. Do you agree with most people?
The strange answer is that most people are wrong and in the above scenario, the chances of the gun firing on the second trigger pull have actually reduced to 25%.
“How can that be?” I hear you say. “Nothing has changed from the first trigger sequence where you showed the chances to be 40% of the gun firing on the second trigger pull”.
The mystery here is that things have changed in that we now know that the 2 bullets are in adjacent chambers in the revolver! We didn’t know that before.
Cells of Drug Probability
There are several problems in the A Hen and a Half (Ahaah) book that defy initial intuition, such as Definitely Not a Birthday Party and be baffled by The Spider and the Fly.
Here is another probability situation, where the solution just doesn’t make sense on first sight and here it is.
A medium sized prison holds 2,000 in-mates, 5% of whom are persistent drug users. The strict prison governor is aware of the problem and wants to totally stamp out all drug use within his domain.
The jail is provided with 2,000 test kits to enable all 2,000 inmates to be tested. The test kits have an accuracy of 95%
The question is this:
If a random prisoner is tested positive, what is the likelihood of that prisoner being one of the drug users? 95% I hear you say.
Think about this and the true answer can be downloaded below.