On televised poker shows, they often display the probability that each player is likely to win the hand. This leads to games being essentially about the probability of an upset - can the 20% guy pull out the victory with the next card? I guess this makes for some dramatic tension, but it's not terribly useful for understanding poker.
The main reason is that what they display are full information probabilities - if you knew both players' hands, this is what you'd calculate the odds as being.
Of course, the whole point of poker is that you don't know what the other guy is holding, and you're trying to infer it. The question of how exactly you infer it, from the cards on the board and the way he's betting, is the entire art and science of the game.
The most scientific (or at least probabilistic) part is knowing your chances of winning given only the cards in your hand. If you're only going to display one probability for each player, this is the useful one to understand what the players are actually doing - if there's two players and you hold Ace of Diamonds and 3 of Hearts, what are your odds of winning if all cards are dealt? This would help people understand basic things like why the guy keeps betting if he's only got a 14% chance of winning - he doesn't know that he's only got a 14% chance of winning. He thinks he's got a 54% chance of winning, and doesn't know the other guy is holding a flush.
This number would also be much more useful for helping people learn to play poker better. They'd learn faster what each set of cards implied.
Now, the criticism here is that good poker players will infer much more than the unconditional probabilities based on the flop and how the guy is betting. But if you display both numbers (full information and conditional only on own cards), you'd at least know which way a skillful player was likely to be updating. In other words, he's inferring something between 14% and 54%.
I assume that the TV networks have decided that putting two probabilities on the screen is simply too confusing for the average boob TV audience. But I'm not so sure. Frankly, to watch the game at all, you've got to have some interest in poker, and it is simply impossible to be interested in poker without understanding the rudiments of probability (intuitively, if not formally). The guys who would find this totally confusing probably are never going to watch the show anyway.
I am as skeptical of human nature and ability as the next man, but on this one, I say give viewers the benefit of the doubt and put both full-information and partial information probabilities up.
(As a side note, I initially was going to title this post 'A Modest Proposal For Improving TV Poker', which has a good ring to it. The problem is that Jonathan Swift meant the 'modest' in a sarcastic way, and it adds greatly to the confusion to also use it for truly modest proposals - it's like people who use the Casablanca 'shocked, SHOCKED' line for things that are actually shocking. Don't do it!)