One thing that is often reiterated, is that one should keep up to the pace of the world around. If you stay, you will miss somethings, but there is always an advantage in switching. Consider this classical example called the "Monty Hall" problem in probability theory:
The "Monty Hall" problem relates to a popular TV show, where the participant of the game is allowed to win a Bentley (most expensive car!) or has to be satisfied with a goat (most sort after domestic animal!). The problem is stated as a conversation between Monty and the participant Mr. P
Monty : Welcome, Mr P to this game. Before we begin let me explain. You can see 3 closed doors, behind one of the doors is the most priced Bentley, and behind the others are the useful goats. Choose one and you will be prized with the gift behind the door. You can either go with a shiny, new expensive Bentley or be satisfied with the goat. So are you ready?
Mr. P : Okay let me choose Door __ .
Monty : Okay let us see if the luck supports you. (Pause) Maybe I can give you another chance. I shall open a second door and see if you want to change the choice.
Monty opens a second door and shows that it is a goat.
Monty : Do you wish to stay or switch?
Think that you are Mr. P and write your choice... Stay or Switch!
The chances of winning is more while one switches. The solution to the problem seems to be very counter intuitive. Hence this is regarded as a good example problem for learning elementary probability.
The "Monty Hall" problem relates to a popular TV show, where the participant of the game is allowed to win a Bentley (most expensive car!) or has to be satisfied with a goat (most sort after domestic animal!). The problem is stated as a conversation between Monty and the participant Mr. P
Monty : Welcome, Mr P to this game. Before we begin let me explain. You can see 3 closed doors, behind one of the doors is the most priced Bentley, and behind the others are the useful goats. Choose one and you will be prized with the gift behind the door. You can either go with a shiny, new expensive Bentley or be satisfied with the goat. So are you ready?
Mr. P : Okay let me choose Door __ .
Monty : Okay let us see if the luck supports you. (Pause) Maybe I can give you another chance. I shall open a second door and see if you want to change the choice.
Monty opens a second door and shows that it is a goat.
Monty : Do you wish to stay or switch?
Think that you are Mr. P and write your choice... Stay or Switch!
Figure showing the illustration of the Monty Hall Problem. (Drawn using MetaMoji Note App)
Solution : Mr. P would wish to switch in order to win the Bentley. The reasoning is as follows.
We contrast the chance of winning on switching vs the chance of winning on staying with the previous decision
$\mathbb{P} (win|switch) = \mathbb{P}(I = G, III = B) = \frac 23$ and $\mathbb{P} (win|stay) = \mathbb{P}(I = B, III = G) = \frac 13$.
The chances of winning is more while one switches. The solution to the problem seems to be very counter intuitive. Hence this is regarded as a good example problem for learning elementary probability.
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