I don't know enough about this particular contest -- and since it happened in 1984, I'm not going to attempt to find more information on the internet -- but typically the goal isn't to have an
unwinnable contest, just to have it drag on long enough while drawing more and more people into listening.
At the start of the contest, you might have 0.1% of the people in your listening area tuning in for the reading of the serial number, and by the end, 1-5%, before dropping down to, say, 0.2-0.5% after the contest is over. Early on, there's a 1 in a 1000 chance of someone listening having the bill, more or less, so the contest won't end too early, but by the end, you are up to a 1-in-20 chance, so once it gets popular enough, it will end fast enough that people don't get too much into grumbling that your contest is rigged.
Meanwhile, the radio station probably didn't actually put up the $30,000 -- it was probably an advertiser who paid the radio station $40,000 out of their advertising budget to run the contest, or should I say, the Wonder Widget's daily drawing, brought to you by Wonder Widgets, your wonderful source of wonderful widgets. You repeat the advertiser's slogan three or four times as people are tuning in to the station to listen for the day's serial number, and three or four more times throughout the day reminding people to tune in at X o'clock.
There's no need for the contest to be rigged or unwinnable -- it's a bargain for everyone at half the price. The advertiser knows people are listening to their ads, the radio station is getting paid outright as well as getting a boost in their listeners (which gets them more money from other advertisers). And actually paying out the money, eventually, lets you go back to that same gold mine again and again.