Suited Blackjacks

[S. Yama]

great formula

Tirle_bj and Arlalik,

P = 8N / 13(52N-1) - this is really very elegant formula.
I don’t recall seeing it anywhere. Bravo!
Do you care to share your train of thoughts on how you came up with it?

I was familiar with Peter Griffin’s formula for blackjack (and suited would be a forth of it) for any subset of cards left:

a * t * (1-2(a-1)*(t-1)/(n-2)/(n-3))/n/(n-1)

where:
a = number of aces,
t = number of tens,
n = cards remaining.

 

[Monkeysystem]

This Kept Me Up All Night

Prob. of getting an Ace = 1/13
Prob. of getting suit Ten to that Ace = 4N/(52N-1)

4N is the number of suited Aces. (52N-1) is the number of remaining cards after removing the first Ace.

You multiply the two probabilities together.
Therefore Prob. of A-T suited starting with the Ace = 4N/13(52N-1)

Prob. of getting a Ten = 4/13
Prob. of getting an Ace suited to that Ten = N/(52N-1)
You multiply the two probabilities together.
Therefore Prob. of T-A suited starting with the Ten = 4N/13(52N-1)

It's a combination, so you have to add the two probabilities together.

So 4N/13(52N-1) + 4N/13(52N-1) = 8N/13(52N-1)

Nice work, guys.

 

[mociferous]

Algebra in the real world

Thanks, Monkeysystem et al... This gives me proof for my high school kids that algebra is helpful in the real world!

 

[tirle_bj]

thank you

Thank you all, guys. Good analysis, Monkeysystem!
Yama, here is our way to that formula:
For Probabilities in Denominator we should have all possible combinations of two cards:
[52N * (52N-1)] / 2 = 26N * (52N-1).
In Numerator we should have total number of combinations with suited BJs.
We have 4 suits, N Aces in each suite, and 4N Ten-value cards in each suite. So total we have: 4 * N * 4N = 16(N^2)
The final probability is [16(N^2)] / [26N * (52N - 1)]
To simplify we just divide numerator and denominator by 2N and get...
8N / 13*(52N -1)
Now for Odds we should just turn this formula up side down and here we go:
13*(52N-1) / 8N.
Again it's always pleasure to share opinions and thoughts between each other here on this wonderful site.