Theoretical probability of winning this game?

Lets say there's a game that involves a 40 card deck without any face cards. And to win this game the player has to pick 4 cards and if they pick a 4 card hand of all one suit they win ex.(2 of hearts, 5 of hearts, 7 of hearts, 9 of hearts).
The probability of this would be 4(10c4/40c4) = 840/91390.
And lets say another way of winning this game would be to get a hand of all red cards. ex.(6 hearts, 4 diamonds, 3 hearts, 8 diamonds).
The probability of getting this hand would be (20c4)/(40c4)= 4845/91390.
Since the 2 ways of winning overlap each other, you could get the same hand from both calculations so what should I minus the equation of P(4 red cards) by to find the probability of winning? How should the end equation look like?

Quote:

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Originally Posted by TP123

Lets say there's a game that involves a 40 card deck without any face cards. And to win this game the player has to pick 4 cards and if they pick a 4 card hand of all one suit they win ex.(2 of hearts, 5 of hearts, 7 of hearts, 9 of hearts).
The probability of this would be 4(10c4/40c4) = 840/91390.
And lets say another way of winning this game would be to get a hand of all red cards. ex.(6 hearts, 4 diamonds, 3 hearts, 8 diamonds).
The probability of getting this hand would be (20c4)/(40c4)= 4845/91390.
Since the 2 ways of winning overlap each other, you could get the same hand from both calculations so what should I minus the equation of P(4 red cards) by to find the probability of winning? How should the end equation look like?

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Substract the number of ways that are common to both.

(And obviously to convert into a probability you have to divide your answer by the total number of ways of picking 4 cards without restriction).

Hi TP123,

Use

where
A = drawing 4 cards of one suit and
B = drawing 4 red cards

Do you see what is?

[edit]Beaten to the punch by Mr. F! [/edit]

Thanks mr fantastic and akward (Happy)