So it is easy to see that while the door you are touching has a car behind it is has probability 1/6 , the probability that that the door you are NOT touching has a car behind it is 1/3.
vina-saar, given that there are only two doors left and one of them has a car, shouldn't these probabilities add up to 1 instead of 1/3+1/6=1/2?
Two doors. One with car, the other with goat. No REAL advantage in switching. It is like carrying a bomb on board.
Dileep-saar, that is not true. there are two doors. One has a car and one has a goat. But what is the probability that you have your hand on the door with the goat (apriori, irrespective of what the host does) ? There is another way of thinking about it.
Game 1: There are two contestants, Dilbullah and Vamanullah. None of them know what is behind the closed doors. D picks a door, does not open it yet. V picks one of the doors and opens it. Now irrespective of what V finds
behind the door, is it advantageous to switch ?
Game 2: There is one contestant and one host, Dilbullah and Lalprafessar. D has no idea what is behind closed doors, Lalprafessar knows everything
. D picks a door and does not open it yet. L picks a door and opens it and shows a goat. Now is it advantageous to switch ?
Are Game 1 and Game 2 similar ? (Hint, the are not.) What is the difference ?
Consider a modification of Game 1.
Game 1(b) If V finds a car, is it advantageous to switch ? (if V finds a goat, the game is called off)
Game 1(c) If V finds a goat, is it advantageous to switch ? (if V finds a car, the game is called off) Hint: Does the fact that we have removed those instances that V finds a car and replay the game, have any effect ?
Is Game 2 similar to Game 1(b) or Game 1(c) ? What are the sample spaces (list of possible outcomes) for Game 1(b) and Game 1(c) ? So is it advantageous to switch in Game 2 ?
"YOU carry a bomb on board. TWO independent people carrying a bomb aboard is 1 in 1 billion", said the statistician.
The logical flaw being that the experiment has been influenced and is not the previous experiment anymore. Take for example this experiment.
I have a fair coin, I toss it, what is the probability I will have heads ? It is 1/2
You and I have a fair coin each, we toss both them. What is the probability we will get two heads ? It is 1/4. (Dont drag in Mr Bose here
You and I have fair coins, you cheat and show me a head always. I toss. What is the probability we will get two heads ? It is 1/2 and not 1/4.
The reduction in probability comes only if you toss and not if you cheat. Similarly, replace coin with Bomb. Bomb on the plane is 1 in 24000 (chance of tossing a heads). If you always carry a bomb (always show me heads), probability that someone else has a bomb too is 1 in 24000 and does not change.
The teacher of my 11th grade started the statistics class by telling us that it sometimes gives absurd results. The example she gave was not as funny as the bomb case. She said, statistics show that vehicles hit pedestrians more at the sides of the roads and less at the middle. So, statistically, it is safe to walk in the middle of the road.
That is also not true. This is a subtle wordplay on the conclusion. Let me restate using another example.
Number of people who get killed by snakes is less than number of people who die from complications from common cold. So it is safer to be bitten by a snake than to catch a cold ? Obviously it is safer to be bitten by a snake isnt it ?
The flaw in this line of reasoning is as follows: We should be addressing the question: If you get bitten by a snake
, what is the chance that you die of snakebite ? If you get a cold
, what is the chance you die of a cold ? Now, given a choice of one of the two (getting a cold vs snakebite), wont you pick having a cold over getting bitten by a snake ?
Similarly, in the road experiment, if you walk in the middle of the road
, what is the chance that you die ? If you walk in the side of the road
what is the chance that you die ? Given that you have to walk, is it better to walk on the side or the middle ?
For the walking in the middle of the road part, statistics is of course right and shows lower values for that. But is it safer? Statistics cant answer that question, it can only show the measures. Deciding whether it is safe or not is an inference and a subjective one at that, which statistics cant answer.
That is actually not true. Statistics does measure propensity. If the question is framed properly, statistics does help out. (a) If you compute statistics of one occurrence, and try to answer a totally unrelated question, like any other scientific method, statistics does not help. (b) If you do not factor in all information, like any other scientific method, statistics does not help.
There is a usual misunderstanding of (frequentists vs Bayesian), but both of them agree on the propensity, they simply dont agree if propensity can be explained....