Wednesday, July 31, 2013

Probability and statistics

Saw this in a TV show last night.
Imagine you are a in a TV game show. You are given a chance to choose from three cards of which 2 have picture of a goat on them and one has a picture of a car. Now if you choose the card with the picture of a car then you win that car. If not you win a …goat!
You choose a card. Now what is the chance that you have chosen a car-card? One in three! Let’s open any one of the two cards that you have not chosen. And suppose it turns out to be a goat. Yay! So now what are the chances you have of having chosen the car-card? Fifty-fifty!Okay, now if given a chance would you switch your choice?  No. 

Apparently if you change your choice you increase your chances of winning.

I was able to follow until the NO. But he lost me when he said that we actually increase the chance of picking the car-card when we switch our choice. I don’t know.

Let me try one more time.
Three cards: A    B   C [Now for the sake of clarity know that A is the car card.]
I choose card B. My chances of the car is 1/3
Now open card C. It’s a goat. So my chance of the car is 50-50. 
Will you change your choice from B to A?
No. So my chance of car is still at 50-50?
Um…no! When you change your choice from B to A your chances are 2/3.

But its probability and you still have the 0.33 chance of not winning even if you change!

On further research…
MONTY HALL - The problem is a paradox of the veridical the type, because the correct result (you should switch doors) is at first sight ludicrous, but is nevertheless demonstrably true. It is mathematically closely related to the earlier Three Prisoners problem, and both problems bear some similarity to the much older Bertrand's box paradox.

The most well known statement of the problem is:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

And the understandable solution is 
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

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