Riddle #3

DeepT- That’s the whole point of the ‘riddle’. It wouldn’t be famous if it was intuitive.

Using numbers- If you open your eyes and it says $100, the other side is either $200 or $50, si? And what are the probabilities thereof? 50/50, yes?

Another way to look at it is this:

Assume there are 5 cards in the bag, but they all have the same number printed on them (X on the front, 2X on the back). Assume the experiment is conducted a large number of times (to even out randomness). If you never flip, then your expected outcome is 1.5X. If you always flip, then your expected outcome is 1.35X (1.5X -10%).

(The cards don’t really have to have the same values, either - if they have 5 different values, then assume the mean of those 5 is X - the logic still holds).

Because your logical argument doesn’t account for the fact that winning is twice as good as losing is bad.

Put another way: if the deal is “Pay and flip, and if it’s doubled you keep it, but if it’s halved you get zero,” you would be right. Because then it’s an even-money bet (50% chance to gain the current face value, 50% chance to lose the current face value) and it’s not worth it to pay 10% to flip (becuase flipping is statistically no change, so you’re just giving away 10% for nothing). Similarly, if the deal was winner gets +50% (rather than 100%), loser gets -50%, the flipping fee is lost money.

But that’s not the rule. The rule is +100% vs -50%.

More on this problem here:

Here’s an interesting expression of it (minus the 10% swap fee, and expressing things in terms of the amounts in two separate envelopes, one holding twice the sum of the other)

The following arguments lead to conflicting conclusions:

  1. Let the amount in the envelope you chose be A. Then by swapping, if you gain you gain A but if you lose you lose A/2. So the amount you might gain is strictly greater than the amount you might lose.
  2. Let the amounts in the envelopes be Y and 2Y. Now by swapping, if you gain you gain Y but if you lose you also lose Y. So the amount you might gain is equal to the amount you might lose.

The paradox above occurs whether you know the value of A/Y (i.e. $53) or not…

You flip. The intuitive explanation is this:

Assume card is $200. For now, ignore that 10% premium you pay.

The only choices are you get $100 or you get $400. forget the $200, ignore that. Now you have a 50% chance of either getting $100 or $400. Intuitively you will recognize that taking the chance is better because the favored outcome is 4 times better than the unfavored.

$50 + $200 = $250. Substract $20 premium, that’s $230 average.

$80 or $380. Average is (380+80)/2 = $230.


Those of you who say “flip” are the ones who keep the financial services sector in business.

The person who takes the money he sees and never flips comes out ahead of anyone who flips even once, assuming a perfect distribution and infinite number of tries.

It is absolutely crystal clear that a policy of flipping can only reduce your expected winnings, if there is a fee to flip.

You have to look at it from the point of view of the game operator. When a person draws a card, the maximum he can win (whatever is on the high side of the card) is already determined. Say the high card is 2X. Depending on which side the player puts down, by flipping he can either get 1.9X, or 0.9X. The player will either double a small number, or lose half of a large number. The player can’t see 2X, flip it, and have a 50% chance of getting 3.8X.

To see it from the player’s point of view, imagine you play the game a long time. While it’s true that for any given number you see, there appears to be a 50% chance you can go double or halves by flipping. However, on average, the numbers where flipping will help you will tend to be small numbers, and the numbers where flipping will hurt you will tend to be large numbers. If you think about it, the numbers where flipping will hurt you are twice as large on average as the numbers where flipping will help you.

Well, that depends on what your goals are. If you want to maximize your winnings, you don’t flip, due to the price of flipping. If the chance of getting $100 dollars is more tempting than losing $30 or so, you flip.

Yeah, this is the ‘pull the goalie in the last 60 seconds’ phenomenon.

I dont think I would flip, since instead I wondered why an eccentric multi-billionaire would have such low values printed on the cards.

This is the explanation FTW. Perform the test a million times and look at how you end up. The same type of thing will prove that in a Monty Hall game, you always take the door he offers you.

In the case originally presented by Phil, you stand to win more money on average if you flip. My work:

Scenario 1: Do not flip.
Expected amount: $53 * 100% = $53

Scenario 2: Flip.
Expected amount:
26.5 * 50% (half the time the number on the back will be smaller than the original)

  • 106 * 50% (half the time the back number will be larger)
  • 5.3 * 100% (you always pay the 10% premium)

$60.95 total

Without getting into marginal utility or other needless complications, scenario 2 has a higher expected value than scenario 1. Flip.

It’s pretty easy to visualize the Monty Hall game if you imagine a lot more than 3 doors. I’m trying to find something similar for this.

I can’t get my head to mesh the player’s reasoning with the game operators POV. If I were playing blackjack, I can bet all my money for even odds to double up. In the flip game, if I’m staring at a card, I only need to bet 1/2 my money for even odds to double up. But it doesn’t matter if I flip?

The exected outcome is the same even if you use cards with 1000X and 1x. So it’s 500x. But now if I’m starign down a card I can can bet (virtually) my whole stack for a 50/50 chance at 1000 times my money. If there’s any cost to flipping, I need to not flip to maximize my winnings?

Yep, I’ve done the same thing and got 60.95, which is a good 7.95 more then if you don’t flip in the best case. Of course probability only works well if you do it many many times, so the better question is…

Do You Feel Lucky? Well, Do Ya, Punk?

Another variant of this problem, paraphrased from the wikipedia page:

Two envelopes sit before you on a table, each with a check in them for a sum of money. You can only have one, and you may freely swap. One sum is twice the other. Your hand reaches for the left envelope and you pull it towards you, about to open it. Then you think "If I switch to the other (right-side) envelope, I have a 50% chance of getting double what’s in this envelope. The expectiation is
(.5 x (Left Envelope Check) + 2 x (Left Envelope Check)) / 2 = 1.25 x (Left Envelope Check)

Ahh… I get a +25% expected value by swapping. (In this case there is no cost to swap).

Being no fool, and not wanting to pass up the +25% E.V., I push the left envelope back, and grab the right one. But before I open it, I think "Hmm, the envelope on the left has a 50% chance of having double what this envelope has…

And you swap envelopes endlessly without opening them…

And for that matter, the envelope swapping in the above problem gets even more frenetic if you are told that one of the envelopes has 1000 times the value of whatever’s in the other envelope…

The cases where flips will help you tend to be small bets. The cases where flips will hurt you tend to be big bets. The fact that you don’t know what a big bet or small bet is, doesn’t change that.

I do admit it’s tricky though. I can’t put my finger on why exactly the player’s intuition is wrong. The player’s perspective does seem inconstistent though, and the game operator’s perspective IS consistent.

There is nothing in Phil’s original scenario that indicates this. What evidence can you provide that this is true?

It’s because the number on one side is double the number on the other. The argument is, no matter what the numbers are (whether he has cards that say $50/$100 or $10,000/$20,000), the “losing” bets (ones where you pay and flip the card and discover a lower number) are always for high numbers ($100 or $20,000) while the winning bets are always for lower numbers ($50 or $10,000). The cards are sort of self-correcting, because it’s not like he says “Draw any card and for $10 I’ll let you flip a coin and double the card if it’s heads”. You can only double the cards where you already drew the low value. If you try to double a card where you drew the high value, you automatically lose.

I’m not sure I actually buy this argument, though, since the real-money risk is the same either way – on a $50 card, you can either win $50 (by drawing the 50 and flipping it) or lose $50 (by drawing the 100 and flipping it). Which makes me think that my earlier post was wrong – you don’t flip. The real money at stake is always the same, so the fee is lost money. Who knows.

To belly up to the bar with my own “run it a million times”, I did.


// flip.php
// by me

$youget = 0;
// how much you walk away with

$facetotal = 0;
// total of all facing amounts (no flip sides)

$flipcount = 0;
// how many times you flip

$plays = 100000;
// how many times you play

for( $idx = 0; $idx < $plays; $idx++ ) {

   $face = rand( 10, 50 );   
   // the number with which you start is an int between 10 and 50  

   $halve = rand( 0, 1 );
   if( $halve )
      $flip = $face / 2;
      $flip = $face * 2;
   // the flip side is either half or twice the facing side, 50/50  

   $flip = $flip - ( $face * 0.10 );
   // include the cost if you flip                     

   if( rand( 0, 1 ) ) {  
      // you take the face card
      $youget += $face;
   } else {
      // you take the flip side
      $youget += $flip;
   $facetotal += $face;


$percent = sprintf( "%01.4f", ( $youget / $facetotal ) );
$facetotal = sprintf( "%0.1f", $facetotal );
$youget = sprintf( "%0.1f", $youget );

echo "<br>
echo "you played $plays times.<br>
echo "you flipped $flipcount times.<br><br>
echo "total facing amount:   $facetotal<br>
echo "total you took home: $youget<br><br>
echo "you won $percent of the total.<br><br>
echo "<br>


Some example outputs:

So flipping randomly nets you just above a 7% gain over always keeping the face card.

So by always taking the flip side, you gain about 14% over always keeping the face card, even when including the 10% fee.

The lesson: luck has nothing to do with it. Flip.