Three Slips of Paper

May 28, 2016

RAY: This is from Norm Leyden from Franktown, Colorado.

Three different numbers are chosen at random, and one is written on each of three slips of paper. The slips are then placed face down on the table. The objective is to choose the slip upon which is written the largest number.

Here are the rules: You can turn over any slip of paper and look at the amount written on it. If for any reason you think this is the largest, you're done; you keep it. Otherwise you discard it and turn over a second slip. Again, if you think this is the one with the biggest number, you keep that one and the game is over. If you don't, you discard that one too.

TOM: And you're stuck with the third. I get it.

RAY: The chance of getting the highest number is one in three. Or is it? Is there a strategy by which you can improve the odds?

Answer:

Ray: Well, it turns out there is a way to improve the odds—and leave it to our pal Vinnie to figure out how to do it. Vinnie's strategy changes the odds to one in two. Here's how he does it: First, he picks one of the three slips of paper at random and looks at the number. No matter what the number is, he throws the slip of paper away. But he remembers that number. If the second slip he chooses has a higher number than the first, he sticks with that one. If the number on the second slip is lower than the first number, he goes on to the third slip.

Here's an example. Let's say for the sake of simplicity that the three slips are numbered 1000, 500, and 10.

1000

Let's say Vinnie picks the slip with the 1000. We know he can't possibly win because, according to his rules, he's going to throw that slip out. No matter what he does he loses, whether he picks 500 next or 10. So, Vinnie loses—twice.

Now, let's look at what happens if Vinnie starts with the slip with the 500 on it. If he picks the 10 next, according to his rules, he throws that slip away and goes to the 1000.

Tommy: Whopee! He wins.

Ray: Right. And if Vinnie picks the 1000 next, he wins again!

500-1

Finally, if he picks up the slip with the 10 on it first, he'll do, what?

Tommy: Throw it out. Those are his rules.

10-1

Ray: Right. And if he should be unfortunate enough to pick up the one that says 500 next, he's going to keep it and he's going to lose. However, if his second choice is not the 500 one but the 1000 one, he's gonna keep that slip—and he'll win.

10-2

If you look at all six scenarios, Tommy will win one in three times, while Vinnie will win three times out of six.

Tommy: That's almost half!

Ray: In some countries.

Confused? Here's more...

For those of you still don't believe Ray, here's a chart that demonstrates the possible outcomes, using three unequal numbers. Column A is your first choice, which Vinnie says you'll always toss. B is your second choice, and C is the third number, which you may or may not get to. In this example, we've randomly selected the numbers 2, 14, and 33, but Vinnie's technique works, no matter what the values are for A, B, and C.

So, here are the six possible outcomes:

chart