For years at work, our activities committee (fondly referred to as “The Fun Club”) has held a 50/50 raffle each Friday. A few months back they decided to mix things up a bit by replacing that with something called “Chase the Ace.”
The rules are pretty simple:
- Tickets are a dollar a piece.
- A single ticket is drawn each Friday.
- The ticket holder draws a card from a deck.
- If it’s an ace, you win half the pot. If not, all the money rolls over to the next week.
- Rinse and repeat until an ace is drawn.
This all sounded fine and good until weeks and weeks passed with nobody drawing an ace. In fact, we got up to 15, 16, 17 weeks… and still no ace! Seriously? Wouldn’t the odds of that happening be infinitesimal? Were we somehow being hoodwinked by some nefarious faction in the Fun Club?
With some rusty high school probability theory rattling around in our brains and Excel just a click away, a couple of us computer nerds went to work to find out just how unlikely this really was. We set up a simple spreadsheet with formulas to calculate the odds of pulling an ace up to and including 49 pulls (the worst possible outcome).
First let’s look at the odds of pulling an ace for each individual week.
With a full deck, the first week odds of not pulling an ace are 48/52 or 12/13 or 92.3%. So, of course, the first week your odds are pretty slim. That makes sense. The second week, the odds aren’t much better: 47/51 or 92.1%. It turns out it takes quite a few weeks before the odds of any given week reach some reasonable chance. In fact, to get to a 50/50 chance, you’d have to wait until there are only 8 cards left in the deck!
But those are the week-to-week odds. We all know that if you flip a coin 99 times and get heads each time, the odds of getting heads on flip 100 is still 50/50. But it would be crazy unlikely to get that far in without flipping a single tails. Similarly, wouldn’t compounding a long series of ace “no pulls” escalate in our favor pretty quickly?
Here’s the graph for the compounded probability week-to-week. The blue line is the one we’re interested in. At week 17, the odds are about 1/5 (20%) that no ace would have been drawn to that point. Yes, it’s less likely that this would happen, but it’s still within the realm of believability and certainly not infinitesimal.
The blue line shows the odds if the card pulled each week is not replaced in the deck. For comparison, the green line shows the odds if the card was being drawn from a full deck each week.
So with that, the nerds went back to work, the Fun Club was cleared of any suspicion of hoodwinking, and the pot of cash kept growing.