Originally Posted by Samango
I'm not familiar with that notation Blue. What is C in 50C3, and I'm surprised that you can divide one resultant number by the other to arrive at the result ( because the usual way that I'm familiar with, would be three separate calculations for each card on the flop, multiplied together) Can you show me the workings of the way you do it?
Yea multiplying together works fine in this scenario but can get impossible when you do things that aren't this simple, hence me using the notation I did as that's what I usually use. The C is the the notation used for combinations in nCr, it's a button on a calculator that saves you having to put into your calculator n!/(r!(n-r!)). It's what is used to find the number of combinations when order doesn't matter.
Basically if you think about it it makes sense. If the total number of combos that make your flush can be represented with 11C3 so in total there will be 165 combos of flushes that you can flop. then the total different flops that are possible is 50C3 which is 19600. so to work out the chance of flopping one of the 165 combos from a total of 19600 combos you just divide one by the other. It's just the same as if you have 20 apples in a basket and 5 are red the chance of choosing a red apple is total red/total apples. Then if you want two reds apples in a row the way you calculate the chance is 5C2 (the way you can choose two red apples from 5). Then you would divide this number by the total ways you can choose two apples (20C2)
So the working is basically 11C3/50C3 for the probability, that can be written as (11!/3!8!)/(50!/3!47!). Now if you simplify this it becomes (11x 10x9/3x2x1)/(50x49x49/3x2x1). Which can be simplified to 11x10x9/50x49x48 which is the form you stated of just multiplying all the individual odds together. This would just be the correct way to represent it and can be used for more complex things which can't just be thought of in the sense of multiplying probabilites together.
11!/3!8! can be sinplified to 11x 10x9/3x2x1 because 11x10x9x8x7x6x5x4x3x2x1
). The underlined parts cancel out to give the answer 11x 10x9/3x2x1 Apply the same logic to 50C3.
Hopefully I've explained that so you understand it.