I was playing poker with a friend last night when a question occured to us. I had a two Jacks and the flop came out: King Queen and 4.
So, suddenly my pocket Jacks are not so great, unless another Jack came on the turn/river.
I was thinking to myself, "what are the odds another Jack" comes out and pondering how to apply that. I already know that 5 cards are known (my 2 Jacks and K, Q, 4) and 3 cards are burned/in my friend's hand.
It occurred to me that I should apply Baye's rule here. Instead of "What are the odds another Jack" comes out, it should be:
"What is the probability another Jack will come out given that 5 cards will be/already is burned"
I'm a bit rusty with my probability so I wanted to ask this forum is anyone can help show me how to apply Baye's rule here. (Or if Baye's rule isn't the right way to calculate this, what is?)
The three cards in your friend's hand and burned don't matter unless the betting tells you something about the chance your friend has a Jack. On the turn, you have two favorable outcomes out of $47$ because you know five of the cards. Assuming you don't get a Jack on the river the chance you have two favorable out of $46$ cards for the river. The chance you get no Jack is then $\frac {45}{47}\cdot \frac {44}{46}$ so the chance you get at least one Jack (wouldn't two be nice?) is $1-\frac {45}{47}\cdot \frac {44}{46}=\frac {91}{1081} \approx 8.4\%$