Most people get the probability of a straight or a flush wrong because they consider the act of determining the 7 cards up as a single event. This is an assumption made by a lot of people, and it is wrong. Most people, heard it from a guy who knows a guy who plays in Vegas and he says it is rare. Well that guy who plays in Vegas is wrong.
The fact is, if you count up the 5 cards you need for a straight or flush and divide it by the total deck size, then you are assuming that it all happens in one event. That is not the correct way to analyse it.
The true probability requires you to add the consecutive probabilities of several events together. When the exact event could happen on multiple attempts ( like getting the right cards on the turn, and the river ) then you need to add the probabilities together. That is a mathematical fact.
All these events are mutually exclusive because the cards can only occupy seven slots. Either you get the Ace of spades in your hand or you get a two of diamonds. If that is so, then you need to consider that the probabilities of all cards adds to one after each event.
The error people make is that they don't account for the sequential events.
If you want to know how likely two mutually exclusive and yet coincident events A and B are at the same time it looks like this: P(A and B) = P (A) * P(B). But if we have different events, one that occurs after the first one, then the probability is different. If we have two possible favourable events that don't occur at the same time, A and then B, then that probability is added, not multiplied. P(A or B) = P(A) + P(B). First A happens and then B could happen.
For example, you want to know what is the probability of a second Ace on the flop when you have one in your hand then the total probability is computed for the SINGLE EVENT P(Ace in flop). Assuming no one else gets an ace for 9 players and an ace isn't burned then the flop is one event then
P(Ace in flop) = 3/34 = 0.088235294.
But if you want to know what the odds are and you have several events consecutive then you need to compute the odds of each event and add them.
Let's say you have an Ace in your hand and none show up in the flop, then you have the turn and the river events to pick up that ace. Assume you have an ace, you are playing 8 other people, assume that the aces stay in the deck and aren't burned, and there are 3 remaining in the deck after the flop.
P(Ace) = P(Ace on turn) + P(Ace on river) = 3/30 + 3/28 = 0.1 + 0.107142857 = 0.207142857 or about 20% chance. You have at most a 20% chance of getting that Ace pair best case scenario.
You might get triple aces, one on the turn and one the river, but that is a little harder P(AAA) = P(Ace on turn) + P(Ace on river) = 3/30 + 2/28 = 0.1 + 0.071428571 = 0.171428571.
The odds of flushes and straights are very low when you consider the entire chances for one hand and the flop. But when you play with nine people, there are nine chances you will to get two straight cards (the start of a straight) or flush ( two of the suit) with your hand or in the other 8 hands at the table. Consider the odds at that point with 52 cards
2 2 2 2 2 2 2 2 2 18
-- + -- + -- + -- + -- + -- + -- + -- + -- = ---
52 52 52 52 52 52 52 52 52 52
= 0.346153846
People think that straights and flushes are low probability, but those combinations occur about one in three hands. That's quite a lot.
Most people calculate assuming 52 cards in all for this chance of a straight or flush, but it's only convenient for the first deal. It is a wrong approximation. The card count changes and therefore the probability changes as cards are removed. The trick is that as the card count reduces, the probability of one cards goes up. Sounds odd, but let's continue. Let's calculate the best case scenario for a flush.
Then you bet on those before the flop. The Flop is a separate event.
So you got your two cards and now there are 16 other cards removed. Let's say you have two hearts, Ace of hearts and 2 of hearts. You have 2 of 13 possible hearts. You want 5 of them. Of course some of them will go into other hands. If you make the assumption that hearts are 25% of the cards and there are 16 cards out there, then perhaps on average 4 cards dealt out are hearts in other people hands. The worst case is those cards are together in other hands. But since you have the Ace, and you are risking a flush, then you feel confident you have the best one out there. Of course, if the flop comes up 3 low clubs, you are going to run for the door and fold you hand. But let's say you want to know how good your flush chances are before the flop. Let's assume that the 4 hearts are in other hands. That leaves 7 more hearts. Essentially half of them are still available before the flop assuming random distribution and hearts are 25% of the deck. There will be times when there are no hearts left and there will be times when all hearts are left. But since you can't peak under other peoples hands, they frown on that I know, you need to make some reasoned assumptions.
Now there are 52 - 18 = 34 cards left in the deck. This is the new denominator for all probability. There are 34 cards remaining and you need the 3 of them to be hearts. There are 7 left outstanding. Let's say two hearts do come up on the flop, the eight and nine of hearts. Then 4 cards were used for the flop but the probability was
7
-- = 0.205882353
34
There might have been a heart as the burn card, but since the hearts occupy only 25% of the total cards that will happen on average 1 in 4 times. Let's assume that isn't the case. There are now 5 hearts left for the turn and the river.
There are now 30 cards remaining for the turn card. This is the new denominator for all probability. The turn takes 2 cards and you need one heart and it doesn't matter which. Again, there might have been a heart as the burn card, but since the hearts occupy only 25% of the total cards that will happen on average 1 in 4 times. Let's assume that isn't the case again.
Your probability of getting a heart is now
5
__ = 0.166666667
30
There are now 28 cards remaining for the river card. This is the new denominator for all probability. The river takes 2 cards and you need one heart and it doesn't matter which. Again, there might have been a heart as the burn card, but since the hearts occupy only 25% of the total cards that will happen on average 1 in 4 times. Let's assume that isn't the case again.
If you didn't get a heart on the turn, then you have one more chance at it. Assuming there are still 5 left your probability of getting a heart is now
5
-- = 0.178571429
28
So your total chances, with four hearts after the flop, are
0.166666667 + 0.178571429 = 0.345238096
Compare the chance of getting a flush (when you have four suited cards after the flop) with getting a second ace on the turn or the river.
Chances of getting an Ace = 0.207142857
Chances of getting a Flush = 0.345238096
Of course, this is predicting random events, so there is no guarantee that you will get the right cards when you want them. But don't underestimate just how likely a flush is.
Straights are a little nicer because you have 4 suits to play with. If you have 78 and on the board is 56 after the flop, then for the turn and the river you need either a 4 or a 9 to complete the straight. That is 8 cards left to get a straight rather than 7 above for a flush.
8/30 + 8/28 = 0.266666667 + 0.285714286 = 0.552380953
All these odds calculated are the best case scenario, they will almost never happen this well. In fact, the probability drops away when you start taking the right cards out in burn cards and in other player's hands. But it is important to stress that these are all likely events and you must consider the sequential nature of events to get the right odds.
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