Congratulations, you have been selected from among the audience to compete in the fabulous Blindfold Bingo challenge! (cue annoying, yet catchy theme music.)
Here you see our perfectly ordinary Bingo machine:
Well, almost ordinary! In addition to having just one output tube, it also has two input tubes. (click the image to see the full schematic)
Let's recap the rules: (cue near-silent suspense-building music)
There are 75 White bingo balls. One of them has been randomly preselected to be the ODDBALL. The balls are numbered, so they can be distinguished by looking at them, but not otherwise. All the white balls, including the oddball, are shuffled, and placed into the white input tube in a random order. 75 balls, incidentally, is the maximum number of balls an input tube can contain.
Then there's the yellow input tube filled with 75 yellow bingo balls, indistinguishable from the white balls except by looking.
The bingo cage is initially empty. (music fades out)
You are allowed to use these two actions, as many times you want, in any order, and there's no time limit:
1: choose an input tube, and drop one ball from its bottom end into the bingo cage. The balls will come out of the input tube in the same order that they were put in. (Actions A and B in the schematic)
2: shuffle the balls that are in the cage, randomly pick one, and place it into the input tube of your choice at the top end of that tube. You can only place a ball into a tube that has 74 or fewer balls in it. (Actions C and D in the schematic)
After you are done, all the balls must be back in the input tubes, and if the ODDBALL is now in the YELLOW INPUT TUBE, you win the car! (cue over-the-top sound effects, ridiculous camerawork and scantily clad gentlemen making the cheapo car seem outright luxurious)
Now, let's check your blindfold.. yes, good .. aaaaand GO! (cue lights dimming, dramatically suspenseful music..)
Daily Hint 0:
@TwoBitOperation has posted a strategy that gives better than 50% chance of winning. However, a strategy that wins more than 60% of the time does exist.
Daily hint 1:
@noedne has posted a strategy that gives better than 60% chance of winning. However, a strategy that wins more than 70% of the time does exist.
Daily hint 2:
As of yet, no-one has posted a strategy that gives better than 70% chance of winning. However, a strategy that wins more than 80% of the time does exist.
Daily hint 3:
As of yet, no-one has posted a strategy that gives better than 80% chance of winning. However, a strategy that wins more than 90% of the time does exist.
Daily hint 4:
The intended solution wins about 93% of the time. In a 10000 run simulation, it got, on average, about 70 white balls (70.12) into the yellow tube, and vice versa.