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j1mathman wrote:so that would be
1* [(1/3)^5] *2* [(1/3)^2] *3
to get 5 same 2 different. 0.27 %, so 2 sets with 7 is 99.7%?
Fewnix wrote:j1mathman wrote:so that would be
1* [(1/3)^5] *2* [(1/3)^2] *3
to get 5 same 2 different. 0.27 %, so 2 sets with 7 is 99.7%?
Of course my post doesn't really address the question asked odds of double cashing with 6 cards got the math on that? ?
TheFlashPoint wrote:This is an ideal problem for a trinomial expansion!
There are 3^6 possibilities. Each digit in the trinomial expansion of 3^6 represents a different color partitioning.
1
6 6
15 30 15
20 60 60 20
15 60 90 60 15
6 30 60 60 30 6
1 6 15 20 15 6 1
Sum those numbers, and you should get 3^6, or 729 different possibilities.
Because of color symmetry and without loss of generality, we can consider 1/3 of the solutions out of 1/3 of the possibilities.
1
6 6
15 30 15
20 60 60
30 (One third of the 90 instances where there are two of each color)
So which outcomes represent two sets? Our proof is by exhaustion:
1: All Colors the same. Yes!
6 + 6: 5 of one color, 1 of another. No!
15 + 15: 4 of one color, two of another. No!
30: 4 of one color, one of each other color. Yes!
20: 3 of one color, 3 of another. Yes!
60 + 60: 3 of one color, 2 of another, 1 of a third color. No!
30: 2 of each color. Yes!
Summing the Yeses, we get 1 + 30 + 20 + 30 = 81. Out of 729 / 3 = 243 possibilities.
The odds are 81/243 or one in three.
j1mathman wrote:can you do odds of double cashing with 7 cards? I calculated it as being above 99%, but I'm not sure that is right.
j1mathman wrote:we calculated it for 7 and it was actually about 95%.
laughingcavalier wrote:j1mathman wrote:we calculated it for 7 and it was actually about 95%.
I am famously not very good at maths or even counting, but I understand the truth lies between your two estimates. In fact, as per a post higher up, assuming you don't make a mistake cashing your first set and cash the wrong colour cads, you have the same chance of getting 2 sets with 7 cards as you do getting one set with 4 cards. The odds on that is a bit higher than 2/3 (which is the odds of getting a set with 4 cards if you don't have a set with 3 cards) but a lot lower than 94% or 97%.
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