linguistics and probability

I was watching a friend of mine playing poker online. At the flop, he had a straight draw (not open ended) where if he got a seven he would have caught a straight. One of the players raised and he called. Jack on the turn, the other player raises again and he call. He managed to catch a seven on the river and won a lot of money on that hand. I yelled at him, explaining that his chances of winning that hand were at around seven percent. “Nonsense”, he says saying that his chances were at fifty percent. He said that he needed to catch a seven on the turn or the river. He says that by the time the river comes, he would either catch the seven, or he wouldn’t, one in two, or fifty percent. Now I know his chances were not at fifty percent (he does too, he was just messing with me), but I couldn’t find the words to use to make my argument true. Can I get some help?

You can always carve a situation up into as many or as few possibilities as you like. But there’s no rule saying each is equally probable. People often assume “If there are N possibilities, then each has probability 1/N”, but this isn’t true; there’s no rule of the mathematical theory of probability saying this has to be true. It just happens if you have some reason to postulate that each if the possibilities is equiprobable; in your case, there was no reason to do so.

There’s only really one rule of the mathematical theory of probability: if out of P1, P2, P3, …, it must be the case that one and only one of them happens, then their probabilities add up to 1. Nothing beyond this rule is forced upon you.

By your friend’s logic if you buy one lottery ticket you have a 50% chance of winning since the ticket either wins or it doesn’t.

Now suppose you buy a thousand tickets. Do you still have the exact same 50% chance of winning since you either win or lose?

Obviously that makes no sense at all.

He is confusing the number of possible outcomes with the likelihood of each outcome. Those are not the same thing.

To take your friend’s “logic” to its extreme, consider this: Tomorrow, the sun may rise, or it may not. Two possibilities. Therefore, according to him, there’s only a 50% chance that the sun will rise?

In your example, you didn’t mention how much the opponent bet. If it was sufficiently small with respect to pot then your friend was correct to call.

The probability of filling a gut-shot (inside) straight draw is about 8.5% on any given street (4 cards out of 47). To get the odds of making it by the River (on either street) calculate the conditional probability of not making it and subtract from 1. So 91.5% times 91.3% (odds changed slightly with one less card in the deck) is 83.5%. That’s the probability he won’t fill his straight. The probability he will are 16.5%.

The easy way to calculate approximate post flop odds is to determine your outs (which were four in your example), multiply by 2 and add a percent symbol. So four outs becomes 8%. This is only an approximation because this rule of thumb uses 50 for the number of remaining cards rather than 47 or 46.

There are also Implied Odds which make for a great excuse for whenever someone makes a stupid call. Implied odds are the expected additional return on your investment for the times when you do hit your hand. Is your hand well disguised or will your opponent realize that you’ve made your hand?

And there is also Fold Equity. Poker rewards aggressiveness because when you bet you have two ways to win. Your opponent can fold or he calls and you have the best hand. It also makes for a great excuse when you acted overly aggressively.

A classic Daily Show clip concerning “50-50 probability.”

You should make a point of inviting your friend to all future poker games.

His friend and he were both joking. They were just curious after the fact how to pinpoint the flaw in the reasoning mocked by the joke.

That is correct.

I think the best way to explain it to your friend is with a deck of cards, and a big pile of money. Keep the money.


I’m either going to win the lottery this Saturday or I’m not. Two mutually exclusive outcomes. Therefore it’s 50-50!

Monty Hall has three doors…

As mentioned above, the dude’s jokingly confusing binary with probable.

{I felt that the thread needed a bit of Linguistics since the term was used in the title.}