Odds of Improving to a given hand not guaranteed?

[ILuvPokaChiks]

I got a question about odds against improving your hand. Let's take a flush draw as an example. The odds against improving to a flush on the turn are 4.2:1 as everyone knows. But I just wondered about that so I took a deck of cards and dealt myself two heart cards and two random non-heart cards to my "opponent" and also dealt a flop containing 2 heart cards, so I had the flush draw. Then as several books says: I should improve to a flush once every five tries. So I deal the top card - the turn card, and count to five dealth cards to see if I get a flush. Often I do not. So are he odds not accurate or just an estimate? I always thought they pretty much guarantee that I will get the flush once in five tries?

[cardcounter0]

What are the odds of flipping heads on a coin toss?
2:1. Those odds say if you flip a coin twice, one of the tosses should be heads. So I flipped a coin twice, and got 2 tails. What gives? Don't the odds of 2:1 on a coin toss pretty much guarantee that every other flip will be heads?
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[ILuvPokaChiks]

I see your point. It's just that most books / tv commentary etc almost make it sound like you WILL hit a flush once in five tries. And they say things like: "he should make this call, as he only need to win one out of five times to make it profitable". But to make it profitable then that would have to guarantee that you hit it once out of five?

[BruceZ]

[ QUOTE ]
What are the odds of flipping heads on a coin toss?
2:1.

[/ QUOTE ]

1:1 if it is a fair coin.

[Lost Wages]

Over a large number of trials (turns) you will make a flush one time for every 4.2 times that you do not. Over five trials, the only guarantee is that you will make a flush somewhere between zero times and five times.

Lost Wages

[meep_42]

Isn't it actually 4:1 exactly, as he knows his opponent doesn't have any hearts?

52 - 2 - 3 - 2 = 45 unknown cards
13 - 4 = 9 hearts
36:9 = 4:1

Anyways -- the odds are accurate, no matter the number of opponents in the hand. In a game there will always be 47 cards you haven't seen and 9 flush cards unaccounted for, making 38:9 against making your flush. 38/9 = 4.22, so it's 4.22:1 over an infinite number of trials.

-d

[ILuvPokaChiks]

OK i see, thx for cleaing it up guys!

BTW: I've been wondering, shouldn't the burn cards be taken out of the calculation? I mean the 3 burn cards (I not sure if its 3) they are cards you never get to see.

[aloiz]

You never see the burn cards, but you never see about 40 other cards in the deck as well. If you deal from the bottom of the deck your odds wouldn't change. If you burned 10 cards between the flop and the turn your odds wouldn't change. All the cards are still unseen, and the probability that the burn card is or isn't one of your outs will have no effect on the overall probability of you making your hand.

Here's a simple example to illustrate. There are three cards (As, 2s, 3s, 4s) You are dealt one card face down, and it happens to be the 2s. You want to know the probabilty that the next card flipped up is the As. Well obviously it's 1/3. However if you were to "burn" the top card and flip over the next card that wouldn't change the odds of you flipping over the As. P(As is second card) = P(As is third card). Since the action of burning tells you no additional information about what the burn card is you cannot eliminate any cards, and thus the odds stay the same.

Another way to think about this is as follows: In this example there are two possiblities given that we are dealt the 2s and we burn the top card and flip over the next card. Either we burn the As or we don't burn the As. Probability that we burn the As = 1/3.
Probability that we don't burn the As = 2/3.
So 1/3 of the time after we burn we will have no chance of turning over the As, and 2/3 of the time after we burn we will now turn over the As 1/2 of the time.

So P(We turn over As) = P(As is burned) * P(turn over As given that As was burned) + P(As is not burned) * P(turn over As given that As was not burned) = (1/3 * 0) + (2/3 * 1/2) = 1/3

Hope that helps,
aloiz