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.
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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