I would like some feedback on some of my thinking.
Situation : I pull 3 cards consecutively from a standard 52 deck.
Question 1 : What are the odds of pulling a flush? (all same suit).
i am trying to break it down as simple as i possibly can, my attempt :
Add up all the hearts wich comes up to 13.. So the chance of pulling a flush of hearts becomes: (13/52) * (12/51) * (11/50) =1.29% chance i have 4 suits, so 1,29% *4 = 5.16% chance to pull a flush..
Question 2 : What are the odds of pulling a straight? I use the same logic as above How do i make a straight? A-2-3 , 2-3-4 ,.... J-Q-K
the suit doesn't matter so i can mix any suit. so my math would look like this : TAO = Total amount of
((TAO Aces /52) * (TAO 2's / 51) * (TAO 3's/50)) * 6
the *6 is for the amount of possibilities i can draw the 3 cards. Do this all the way to J-Q-K
((TAO Jacks /52) * (TAO Queens / 51) * (TAO Kings / 50))* 6.
Add these all up to get your total chance for a straight : 3.19% chance Wich from all data i've found is wrong.
Question 3 This is probably the hardest question to answer. I want to calculate the chance of a straight flush. I believe i have 2 ways to calculate this Option 1 : i do the math like above , and end up with a 0.199910% chance to get a royal flush Option 2 : i know the chance of a flush and a straight, and multiply them to come up with the chance for bother events to occure. in my case that would be 0.0516 * 0.0319 = 0.00164604 = 0.164604% chance to draw a royal flush.
It bothers me so much that i can't seem to figure this out. I would love it if some would calmly explain what i'm doing wrong in each question. Please give me some feedback asap :)
Yours sincerely DImitri
There are $\binom{52}3$ equally probable three card "hands"
1 - For each of 4 suits there are $\binom{13}3$ ways choosing card types
2 - Let's assume Q-K-A is a valid straight (if not, multiply my answers for 2. and 3. by $\frac{11}{12}$
There are 12 ways have a straight with $4^3$ ways to choose the suits
3 - There are 12 ways have a straight with 4 ways to choose the suits when we require a straight flush
So 1: $$\frac{4\times \binom{13}3}{\binom{52}3}= 5.176\%$$
2:$$ \frac{12\times 4^3 }{\binom{52}3}= 3.475\%$$
3:$$ \frac{12*4 }{\binom{52}3}= 0.217\%$$