^{ Robert J. Marks March 18, 2024 6 Probability }

# The Two-Sided Lottery Card Paradox and Infinity

_{Assuming the infinite often leads to ridiculous conclusions.}

_{ Robert J. Marks March 18, 2024 6 Probability }

Here is a fascinating conundrum to tickle the brains of nerds. Its resolution tells us something about infinity.

The problem is the two-sided lottery card paradox. The winnings amount on one side of a lottery ticket is known to be double of what is on the other. For example, $5 and $10 could be hidden under the rubs of a card. Or $1 and $2.

You buy a lottery card featuring scratch-off sections on both sides. The amount on either side is revealed only by rubbing the card with the edge of coin or your fingernail. Upon scratching off one side, a monetary value is revealed. You have the option to either retain that amount or flip the card over, scratch off the other side, and be awarded the newly revealed sum.

You buy a ticket. Side 1 is scratched revealing a $40 prize. Should you keep the prize, or flip the card over and claim the amount on the other side? Since one side of the card is double the other side of the card, Side 2 must either be half ($20) or double ($80) the $40.

To flip or not to flip. That is the question.

Let’s reason. Keeping Side 1 gives a winnings of $40. If you flip, the chances of getting $20 or $80 on Side 2 is 50-50. The average of the winnings on Side 2 is therefore ½ ($20 + $80) = $50. Therefore, your winnings, on average, will increase from the $40 on Side 1 to $50 if you choose Side 2. So, the clear answer is to flip the card and take whatever amount appears on Side 2. On average, you will make more money.

This sounds like a reasonable solution until you think more deeply. The reasoning just presented will apply no matter what the amount is on Side 1. Here’s why. Suppose the amount on Side 1 is X $. The amount on Side 2 is then either ½X $ or 2X $ the average of which is 1.25 X$. The amount on Side 2 is thus always 25% more on average than Side 1. Therefore, always flip.

According to this reasoning, you always choose the unknown winnings on Side 2 no matter the value on Side 1. Curiously, then, you don’t even have to look at the value on Side 1. Save scratching time and always flip the card over and scratch there. This always assures a 25% increase in your winnings over the amount shown on Side 1 – whatever it is.

If this sounds like something is missing in our thinking, it is. Before we reveal the resolution of this paradox, give it some thought.

(THINK THINK THINK)

Thought enough? OK. Here’s the resolution.

(SPOILER ALERT!)

The two-sided lottery card paradox has its resolution in the nonexistence of the infinite. Infinity exists as an abstraction in our minds but not in reality. The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.” As I have written previously on *Mind Matters News*, accepting infinity can lead to inane, ludicrous, and nonsensical conclusions. Here are some examples.

- Even when
*Hilbert’s Hotel*has an infinite number of occupied rooms, it never is in need of a “NO VACANCY” sign. If one, two, or an infinite number of new guests arrive at the fully occupied hotel, room assignments can be rearranged to give each of the new guests a room without kicking any of the current residents out of the hotel. Details are HERE. - Infinite time cannot exist. If it did, an autobiographer who takes a month to write about each day of his life could finish his book. This ridiculous conclusion is a compelling argument that the age of the universe must be finite. A more in depth discussion is HERE.
- The assumption of infinity can be used to prove that the number of locations on a line segment is the same as the number of locations within a square. Likewise, the infinite number of locations on a small square is the same as the infinite number of locations on a larger square. This is a weird conclusion born from an assumption of infinity. Click HERE for details.
- Here’s another weird conclusion resulting from an assumption that infinity exists. Randomly generating a sequence of coin flips with heads as one and tails as zero will eventually generate the binary code for all of the books in the US Library of Congress. Read more HERE.
- Some infinities are bigger than others. The infinite number of points on a line segment is bigger than the infinite set of all counting numbers. A more detailed discussion of this weird conclusion is HERE.
- To this list we add the two-sided lottery card paradox.

Yes, the paradox is due to the nonexistence of infinity in reality. An assumption of infinity lies sneakily in the statement of the two-sided lottery card problem. There is no ceiling on the amount of money on Side 1 and Side 2 of the lottery card. The entries could be $20 and $40. Or they could be two trillion and four trillion dollars. Two dollars and two trillion dollars are equally likely. Imagine making such lottery tickets. First you would generate a uniform random number between zero and infinity on one side of the ticket and enter half the amount on the other side. But generating a uniform random number between zero and infinity is not possible. A uniform random number can be chosen between zero and a trillion, but not between zero and infinity.

In practice, the lottery award amount must be bounded – at least softly. A lottery ticket awarding a prize equal to ten times the US National Debt ($345 trillion dollars) but could be possible with the rules given. Even as a practical matter, winning amounts must be capped or, at least, probabilistically diminished.

This resolves the two-sided lottery card paradox. The winnings cannot be made arbitrarily large.

To illustrate, place a $200 cap on winnings. You scratch off Side 1 and see winnings of $150. Side 2 cannot be double this because it would exceed the $200 winning cap. So you claim the $150 knowing it is the maximum of two sides.

What if the winning cap is $200 and Side 1 shows $50. Here its wise to look at Side 2 which could be double that. So take a chance and rub off Side 2 in this case.

So keep the winnings on the first side if the amount is above $100. If below $100 flip the card and claim the amount of Side 2. If these new rules are followed, the winning on average over numerous games is about $94.

Here’s the takeaway. Infinity does not exist in reality. There are not an infinite number of parallel universes, and placing an infinite number of monkeys at keyboards to generate all literature ever written is thought candy.

Assuming the infinite often leads to ridiculous conclusions.