How come you can't figure your chances of hitting by using your outs

[Your Mom]

I realize that it doesn't work to figure your chances of a flush draw by (9/47) + (9/46) after the flop. you have to do 1 - (38/47) - (37/46). I just don't get it why the first doesn't work. I am sure it is fairly obvious but I just feel dumb for not getting it because I have always been very strong in math and logic. Someone help me.

[RiverMel]

Because the two events are not independent. You can use the multiplier only in cases where you're trying to figure out the chances of at least one of two independent events happening. In this case, the probability is conditional.

[ccwhoelse?]

[ QUOTE ]
I realize that it doesn't work to figure your chances of a flush draw by (9/47) + (9/46) after the flop. you have to do 1 - (38/47) - (37/46). I just don't get it why the first doesn't work. I am sure it is fairly obvious but I just feel dumb for not getting it because I have always been very strong in math and logic. Someone help me.

[/ QUOTE ]

i had the same question a few days ago, then i realized why you can't.

first, it's not (38/47) - (37/46) or (9/47) + (9/46)

you have to multiply (38/47)*(37/46). ok so, you can't figure your chances to make the flush by doing (9/47)*(9/46) because the first number (9/47) is the chance your flush card will come on the turn. meaning, when you flop a four flush, your outs are 9. (9/47) implies you hit one of your outs on the turn, giving you the flush. so (9/47)*(9/46) really doesn't mean much. also, if you wanted to know the chance that two cards of your suit would come out on the turn and river after you flopped a four flush it would be (9/47)*(8/46), the chances that the turn and river give a cards of the same suit.

[pudley4]

9/47 = the probability of hitting on the turn.

9/46 = the probability of hitting on the river if you miss on the turn. You won't always miss on the turn (you'll only miss 38/47 times), so you can't just add 9/46, you have to add (9/46)*(38/47) - this is the probability of missing the turn then hitting the river.

There are 4 possibilities:

A - You miss on the turn and the river (38/47 * 37/46)
B - You miss on the turn and hit on the river (38/47 * 9/46)
C - You hit on the turn and miss on the river (9/47 * 38/46)
D - You hit on the turn and hit on the river (9/47 * 8/46)

Notice that B, C, and D show all the possible ways you can win the hand. You can calculate each of these 3 separately, then add them together. A shorter way is to notice that C and D added together equal the probability you hit on the turn. So you can calculate B, then add 9/47. This is the same as the beginning of my post.

Or, you can notice the easiest way - since A shows all the ways you can miss, you can calculate A, then subtract the result from 1 (since all the probabilities added together must equal 1)

[Your Mom]

Thanks, its been a couple of years since I took statistics. I obviously needed a refresher course.

[BruceZ]

You got a couple different methods now from this thread:

1 - (38/47)*(37/46) = 35% (method 1)

9/47 + (38/47)*(9/46) = 35% (method 2)

Here is another one which is similar to what you tried to do originally. On the flop, before the turn and river cards are dealt, the chances of either card being a flush card is the same, 9/47 (not 9/46 since the turn is not dealt yet). If you just take 9/47 + 9/47, this would double count the times both cards are flush cards, so you have to subtract that off:

9/47 + 9/47 - (9/47)*(8/46) = 35% (method 3)

Regardless of which method you prefer, it is imperative that you completely understand all 3 methods as they each represent fundamental principles that are central to probability theory. If you can become equally proficient with all 3 methods, that really goes a long way towards making you very versatile in many different probability calculations. Achieving this level of versatility requires a solid foundation in understanding and applying the concepts of independence, mutually exclusive events, and intersections and unions of events. These concepts are not difficult, but they are somewhat subtle and often not fully appreciated by students. Some problems such as this one can be done easily by several different methods, while some more difficult problems can be done easily by only one of the methods, so it pays to know them all.

[Paul2432]

A good way to think of why simply adding 9/47 + 9/46 does not work is to realize that some of the time that you hit on the river it will not make your hand, because you will have already made your hand on the turn. Hence, you will be over counting some of your outs.

This becomes apparent if you think about a case where you have more than half the deck as outs. Say 30 outs. Your method would give you a chance of making your hand of 30/47 + 30/46 which is greater than one. Obviously, the chance of making your hand cannot be greater than one.

Paul