# Thread: Theoretical probability of winning this game?

1. ## 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?

2. 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?
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).

3. Hi TP123,

Use $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

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

Do you see what $A \cap B$ is?

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

4. Thanks mr fantastic and akward