6/11/18

We've been to Dave & Buster's a few times. While we like the video games and air hockey, we also like the games where you get tickets (electronic tickets in this day and age). I noticed this machine and had to analyze my chances using probability.

As you can see, the ball has to go through 3 gears to make it to the top and win the maximum number of tickets. Each gear has a number of "Up" spaces that will take you to the next level. Each gear also has a number of "Miss" spaces that will cause your ball to not move up, ie. lose the game. Here is a close up of the first gear:

Let's analyze each gear.

- Gear 1: 9 Up, 3 Miss, 12 Total
- Gear 2: 8 Up, 4 Miss, 12 Total
- Gear 3: 6 Up, 6 Miss, 12 Total

First, notice that it gets more difficult to go up levels since there are more Miss slots. Putting it all together, the chance of getting to the top, is 9/12 * 8/12 * 6/12 = 432/1728 = 1/4, or 25%.

In this analysis, we are assuming balls enter gears randomly and the gears rotate freely of each other. I am not quite sure that the latter is true.

Similar machines can be analyzed in a similar way. We can also compute our expected winnings/losses, as well as compare to other ticket machines to see which gives you the most bang (tickets) for your buck (credits/dollars). I think next time we go to Dave & Buster's I will look for other opportunities.

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