From a deck of 52 playing cards, 7 cards are dealt. What are the odds of the following event occuring

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 one 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) = 1573/1029112
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)

Answer 3

To calculate the odds of the event occurring, we'll first find the total number of favorable outcomes and then divide it by the total number of possible outcomes.

1. Total Favorable Outcomes:
First, we need to determine the number of ways to choose 4 cards from one suit. Since each suit has 13 cards, this can be calculated as a combination of 13 cards taken 4 at a time, denoted as C(13, 4). So, the total number of ways to choose 4 cards from one suit is:
C(13, 4) = (13!)/(4!(13-4)!) = 715.

Next, we need to determine the number of ways to choose 3 cards from another suit. Again, this can be calculated as a combination of 13 cards taken 3 at a time, denoted as C(13, 3). So, the total number of ways to choose 3 cards from another suit is:
C(13, 3) = (13!)/(3!(13-3)!) = 286.

Now, we'll multiply the number of ways to choose 4 cards from one suit by the number of ways to choose 3 cards from another suit: 715 * 286 = 204,590.

Therefore, the total number of favorable outcomes is 204,590.

2. Total Possible Outcomes:
To find the total number of possible outcomes, we need to calculate the number of ways to choose 7 cards from a deck of 52 cards. This can be calculated as C(52, 7):
C(52, 7) = (52!)/(7!(52-7)!) = 133,784,560.

Therefore, the total number of possible outcomes is 133,784,560.

3. Calculating the Odds:
To find the odds, we divide the total number of favorable outcomes by the total number of possible outcomes and express it as a ratio.

Odds = Total Favorable Outcomes / Total Possible Outcomes
Odds = 204,590 / 133,784,560

Simplifying the ratio, we get:
Odds = 99 / 16,722,971

So, the odds of drawing 4 cards from one suit and 3 cards from another suit from a deck of 52 playing cards is 99/16,722,971.