Question

From a deck of 52 playing cards, 7 cards are dealt. What are the odds of the following event occuring
--4 From one suit, 3 from another

The probability of the event occurring is also acceptable. But, the answer is 99/16722971 (as far as the odds go

Answer 1

Could be:
HHHHCCC
HHHHSSS
HHHHDDD

Now, each of those can be arranged in 7!/(4!3!) or 35 ways

let' calculate the prob of <b>one</b> of these, HHHHCCC

Prob(HHHHCCC)
= (13/52)(12/51)(11/50)(10/49)(13/48)(12/47)(11/46)
But the same prob is true for all of the 35 cases,
so multiply that by 35 to get

prob(4from one suit, 3 from another) = <b>1573/1029112</b>
so the prob of NOT any of those = 1 - 1573/1029112
= 1027539/1029112

So the odds in favour of 4 from one suit, 3 from another
= 1573 : 1027539

check my arithmetic, I did not get the same answer you gave.

Answer 2

oops, for some reason I read it as
4 Hearts, 3 from another

so it could be
HHHHCCC
HHHHSSS
HHHHDDD

DDDDHHH
DDDDCCC
DDDDSSS

SSSSHHH
SSSSCCC
SSSSDDD

CCCCHHH
CCCCSSS
CCCCDDD

So there are 12 of these and each can be arranged in 35 ways
so to get the prob of one of these 420 cases, multiply
(13/52)(12/51)(11/50)(10/49)(13/48)(12/47)(11/46) by 420

(my calculator is overheating)