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There's a good chance you're going to be looking for a conversation starter this weekend. Since we at VocaLabs are obsessed with statistics, we think there's nothing better than a mathematical brain teaser. Your mileage may vary. I've got two, one basic and the other far from simple. And since these are conversation starters, I'll give you the answers too.

Coin Flip

For the first one, you need to develop a new nervous habit:  flipping coins.  It won't take long before you've gotten three heads in a row or three tails in a row.  At this point, walk up to someone and say, "I just flipped three heads in a row.  What do you think the chances are that the fourth one will also be heads?"

The Monty Hall Problem

On the TV show Let's Make a Deal, the host, Monty Hall, gives away fabulous prizes to people willing to make fools of themselves.  One of his games is to show contestants three doors.  Behind one door is a fabulous prize, such as a brand new washing machine, while behind the other two there are goats.  The contestant picks one door, say door #1.  Monty then opens one of the other doors, say #2, to reveal a goat.  Then he asks the player:  "Do you want to stick with door #1, or do you want to go with door #3?"

What should you do?  Keep in mind that his job is to torment the contestants, so he will not open the chosen door (#1 in this case), nor will he open the door with the prize behind it.

Solution 1

Assuming the coin is fair, flipping it will come up heads half the time and tails half the time.  No ifs, ands, or buts.  People often think that the coin is due for another tail, to keep things even.  But the coin has no knowledge of its past, nor does it have stake in its outcome.

It's true that the chance of four heads in a row is one in sixteen (1/2 to the fourth power), but that is the combined probability of four independent events.  In this case, the first three are already given.  And while four heads in a row may seem special to our pattern-sensitive brains, it's no more special than heads-tails-heads-tails or any other specific outcome sequence.  That's why computers are better than people at choosing lottery numbers:  when people try to be random, they use the predictable pattern of eliminating anything that looks like a pattern, such as two identical digits in a row.

Solution 2

This is a famously counter-intuitive problem which originally stumped many mathematicians.  In fact, on the real game show, Monty Hall opens the door but does not allow contestants to change their answer.  As he wrote to the mathematician who proposed the problem,

Although I am not a student of a statistics problems, I do know that these figures can always be used to one's advantage... And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.

Monty Hall is right:  the extra information is a big advantage.  Consider the three, equally likely possibilities:

1. The prize is behind door #1.  He randomly opens one of the remaining doors, revealing no extra information.  You win only if you stick with door #1.
2. The prize is behind door #2.  He won't open door #1, since you chose it.  And he won't open door #2, since the prize is there.  So his only option is to open door #3, thus indirectly revealing that the prize is behind door #2.
3. The prize is behind door #3.  As above, you have forced his hand and he must open door #2, thus revealing the location of the prize.
   

So to add it up:  one third of the time you win by staying with door #1;  two thirds of the time you win by switching.  It's as if Monty Hall, rather than revealing a door, had said,

I'll make you a deal.  If you give up on door #1, I'll let you have the prize, whether it's behind door #2 or #3.  You only lose if the prize was behind door #1.

Posted by David Leppik