Should you take the $100,000?
The offer
Imagine you're offered a choice. Take $100,000, guaranteed. Or take a shot at $5,000,000, but only if you bowl a perfect game at your local alley within one month. Miss, and you get nothing. The offer is hypothetical, but it's a good way to see what happens to a long shot when you get to try it thousands of times.
What a perfect game takes
A game of bowling has 10 frames. A strike means knocking down all ten pins with your first ball. A perfect game is 12 strikes in a row: one in each of the first nine frames and three in the tenth, which is the most the tenth frame allows. It scores 300, the highest score possible.
So the chance of a perfect game is your strike rate raised to the 12th power. Treat every throw as independent, with the same strike rate each time, and you multiply that rate by itself 12 times.
A top professional strikes on roughly 65% of first throws:
0.6512 ≈ 1 in 176
A casual bowler is closer to 30%:
0.3012 ≈ 1 in 1,880,000
For one game, the casual bowler's odds are worse than one in a million.
What the bet needs to be worth it
The $5,000,000 option is worth your chance of winning times $5,000,000. That's its expected value. For it to match the sure $100,000, you need a 2% chance, because 2% of $5,000,000 is $100,000. A 1 in 1.9 million shot is nowhere near 2%. But you don't get one game. You get a month.
A very generous month
Here's the best case for a casual bowler with a 30% strike rate:
- You bowl every one of the 30 days.
- You spend 8 hours a day sleeping and eating, which leaves 16 hours a day at the lane.
- Each throw takes 20 seconds on average.
- The moment you miss a strike, you restart the game, since there's no point finishing a game that can't be perfect.
That last rule matters most. You miss on the first throw 70% of the time, so most games end right away. A game averages only about 1.43 throws.
Sixteen hours at 20 seconds a throw is 2,880 throws a day, or 86,400 over the month. At about 1.43 throws a game, that's roughly 60,480 games. Each one has a 1 in 1,881,676 chance of being perfect. The chance that none of them is:
(1 − 1/1,881,676)60,480 ≈ 96.8%
So the chance that at least one of them is perfect is about 3.2%.
The answer
At 3.2%, the long shot is worth about 3.2% of $5,000,000, or roughly $158,000. That's about $58,000 more than the guaranteed money, so on expected value alone, the bet wins.
The margin is thinner than it looks. At 16 hours a day and 20 seconds a throw, the bet only beats $100,000 if your strike rate is above about 29%. At 25%, your chance over the month drops to 0.4%. At 40%, it jumps to 58%. Small changes in strike rate swing the answer hard, because the rate gets multiplied by itself 12 times.
Expected value also leaves out what the month costs you. Taking the guaranteed money means you don't spend 30 days in a bowling alley throwing nearly 3,000 balls a day, which probably isn't even physically possible. And even in this best case, there's a 96.8% chance you walk away with nothing.
Try your own numbers
The calculator at the top of this page lets you set your strike rate, how many hours you bowl a day, and how many seconds each throw takes. It shows your chance of a 300 in 30 days and what the $5,000,000 is worth to you. Press "Bowl the month" and it simulates every throw, one square per day, until you roll a perfect game or run out of days.