I ran across an image on the Internet recently that intrigued me. A device titled “Baby Shoes”.

The URL link associated with the image was dead but it looks like an old countertop gambling machine, perhaps from the 1930s or 40s. Apparently you put a nickel in the machine, then it rolls five dice.

If the sum of the dice is 25 you win “Two Packs”. If the sum is 8, you win “Six Packs”, and so on:

Sum of Dice Prize =========================== 25 Two Packs 10 Two Packs 9 Four Packs 8 Six Packs 7 Ten Packs 6 Twenty Packs 5 Forty Packs 30 Forty Packs

In old gambling games like this, the winning items are often disguises for money, to avoid gambling laws. For example, maybe “Six Packs” supposedly means six packs of chewing gum but really means six dollars or six half-dollars or something like that.

I love calculating probabilities so I wondered what the probabilities were for this game. I’ve done problems like this many times, but I know from experience these math problems are extremely tricky and it’s very, very easy to make a mistake.

When I see a problem like this now, my method of choice is to write a simulation program instead of doing the combinatorial mathematics.

Using 1,000,000 simulated plays, my results were:

Frequency of sum = 25: 0.0162 Frequency of sum = 10: 0.0161 Frequency of sum = 9: 0.0090 Frequency of sum = 8: 0.0045 Frequency of sum = 7: 0.0019 Frequency of sum = 6: 0.0006 Frequency of sum = 5: 0.0001 Frequency of sum = 30: 0.0001 Frequency of other sum: 0.9514

As I suspected, the payoffs mostly make sense but aren’t entirely mathematically consistent. For example, The probability of getting a sum = 9 (p = 0.0090) is twice that of getting a sum = 8 (p = 0.0045) but the payoff for getting a sum = 8 isn’t twice as much as getting a sum = 9.

The name “Baby Shoes” is a bit odd, but luckily I love old movies and I’ve watched scenes where a gambler is about to roll a pair of dice and says, “Come on! Baby needs new shoes!” to encourage a good result so he can win and buy his baby a pair of shoes.

Part of my lifelong love affair with mathematics and computers is due to writing simulations programs like this when I was in college. I can still vividly remember writing a Craps dice game simulation during my undergrad days at U. C. Irvine. Baby Shoes is an easy problem but there are many types of problems that just can’t be solved using standard mathematics because they’re too complex, and a simulation is the only feasible approach. For example, imagine a dice game with a set of 100 dice.

*Former first lady Michelle Obama (left) and current first lady Melania Trump (center) shown getting off Air Force One, both wearing tennis shoes. My hunch is both of them have plenty of shoes. I’d say the girl on the right could use a new pair of shoes, geek-appeal of her current pair made of Legos nonwithstanding.*