Monday, September 20, 2010

Thinking Outside the Envelope

Two white envelopesMarilyn vos Savant got my family talking yesterday. Her Parade column featured this question from a reader in Maryland:

Four identical sealed envelopes are on a table. One of them contains a $100 bill. You select an envelope at random and hold it in your hand without opening it.

Two of the three remaining envelopes are then removed and set aside, unopened. you are told that they are empty.

You are given the choice of keeping the envelope you chose or exchanging it for the one on the table.

What should you do? A. Keep your envelope. B. Switch it. or C. It doesn't matter.
The answer was B (switch to the one on the table), with this explanation: "The latter has a 75% chance of containing the $100 bill."

What? my teenager exclaimed. That makes no sense. It's obviously a 50-50 chance that either envelope could have the money.

No, explained her father. If you put your original envelope back on the table and completely mixed the two up, then chose again, that would be a 50-50 chance. But as the story has it, the one on the table has a 75% chance because it now includes all of the chances that used to belong to the removed envelopes.

He also gave an excellent example: Imagine it had been 100 envelopes, of which you picked one, then 98 empty ones were removed. Would you still think it was a 50-50 chance that yours was the full one?

At that point, she admitted the answer and the 75% chance made sense.

But I agree, it's funny how counter-intuitive it seems. Another fine example of the illusions our brains are more than happy to make real.

I wish school math programs spent more time on probability and statistics. We need a citizenry that can understand odds and critically read the many interpretations of data that are published every day.


Becky said...

I still don't understand. Didn't the envelope you picked up also absorb some of the percentage that was left when the 2 empty envelopes were removed? It is essentially just taking away those 2; it shouldn't matter where the other 2 are, should it? Maybe my brain just isn't functioning correctly today.

Becky said...

After Googling a bit I figured it out (thanks to Wikipedia and some math forums). Thanks for the education!

Carmella said...

I would stick with the one I possessed and then growl-lunge for the one on the table too. Then I would increase my odds to 100%.

Kobayashi Maru! (sort of)

Ross said...

Yes, you should switch, but the probabilities that you mentioned are incorrect. After removing all but one envelope, the probability of the envelope on the table containing the cash is 50%. (It's on the table or it's the one you already chose.) BUT, the probability of the cash being in the envelope you picked is just 25%. (you picked one of the four possibles).

Ross said...

Oops. Sorry, I mis-calculated. The odds the the cash envelope is/was on the table is 75% and the odds that you chose the cash envelope is 25%. Add those together to make 100%. If you put yours back on the table and mix them up and re choose then it's 50/50 to again total out to be 100%.

Daughter Number Three said...

Yes, right. I see Marilyn vos Savant reiterated her answer today in Parade, I imagine because she's gotten so many letters arguing with her.