# Chance of King and red card in the same hand

#### KobeT

##### New member
Hi everyone!

I am writing a paper about probability. And i was wondering how I can solve the following questions:

'What is the chance that you get a king and a red card in the same hand during a card game?'

and

'What is the chance that you get a king and a queen in the same hand during a card game?'

every hand contains 2 cards...

Can anyone help me with this?

#### ksdhart2

##### Full Member
In the first scenario, you'll need to break it down into two separate events. Can you see why your scenario is equivalent to the probability of [(Red King) and (Another Red Card)] or [(Black King) and (Any Red Card)]? So, how many cards are in a deck? How many red kings are in the deck? Since there's only one of each card, any card is equally likely. What does that make the probability of drawing a red king? After you've drawn a red king, how many cards total are remaining in the deck? How many red cards? What does that make the probability of drawing one of those red cards? And what does that make the overall probability of drawing a red king then another red card?

Except now we have a problem. This only accounts for the specific scenario where the red king is drawn first. What if the red king is drawn second? If you go through the same train of thought as before, does that change the probability? What does that make the probability of drawing a (red king then red card) or (red card then red king)?

Now let's tackle the scenario where the king is black. We still can draw the king first or second, so we definitely have to do something to account for this, but can you see why we have a shortcut here because the probability is the same whether the black king is drawn first or second? What does that make the probability of drawing a (black king then red card) or (red card then red black king)? Finally, can you put everything together and figure out the overall probability of this scenario happening? If you can figure out the first scenario, the second one will be the same idea, only easier to work with.

#### pka

##### Elite Member
I am writing a paper about probability. And i was wondering how I can solve the following questions:
'What is the chance that you get a king and a red card in the same hand during a card game?'
and 'What is the chance that you get a king and a queen in the same hand during a card game?'
every hand contains 2 cards...
There are $$\displaystyle \binom{52}{2}=1326$$ possible two card hands. ksdhart2 is correct about cases but must careful not to count some pair twice.
Suppose the two cards are a red king and black five. Does that count as a king and a red card?

#### KobeT

##### New member
There are $$\displaystyle \binom{52}{2}=1326$$ possible two card hands. ksdhart2 is correct about cases but must careful not to count some pair twice.
Suppose the two cards are a red king and black five. Does that count as a king and a red card?
Yes it does, so how can I calculate it then?
Sorry, I just find this subject very hard and it is difficult for me to undestand it...

#### pka

##### Elite Member
Yes it does, so how can I
We want to count all the two card hands that contain at least one king and a red card. We must be careful to include all such hands without over-counting. Cases:
1. There are $$\displaystyle (2)(48)=96$$ hands containing a red king and any non-king.
2. There are $$\displaystyle (2)(24)=48$$ hands containing a black king and any red non-king.
3. There are $$\displaystyle 5$$ hands containing two kings not both black.
Do those three cases exhaust all possibles for two card hands that contain at least one king and a red card?
If that is a correct count then the probability is $$\displaystyle \dfrac{149}{1326}\sim 0.1123680241327$$.