Question 459356
<pre><font face = "consolas" size = 2><b>


Find the probability of a prime number under 10 given 
the card is red. (1 is not prime.)

Start with a full deck of cards:

<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824;  K&#9824;  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827;  K&#9827; 

Remove all cards except the ones that are given.
We are given that it is red, so we remove all cards
except the red ones.  So we have only these 26:

<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>

Now we will select the ones that are prime numbers under 10,
which are the 2's, 3's, 5's, and 7's. I'll enclose them in
brackets:

<font color = "red">
 
A&#9829;  [2&#9829;] [3&#9829;]  4&#9829;  [5&#9829;]  6&#9829;  [7&#9829;]  8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;  [2&#9830;] [3&#9830;]  4&#9830;  [5&#9830;]  6&#9830;  [7&#9830;]   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>

That's 8 out of 26 or 8/26 which reduces to 4/13

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Find the probability of a King, given that the card is not a heart.

Start with a full deck:
<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824;  K&#9824;  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827;  K&#9827; 

Remove all cards except the ones that are given.
We are given that it is not a heart, so we remove all 
the hearts.  So we have only these 39:

<font color = "red">
  
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824;  K&#9824;  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827;  K&#9827; 

I'll enclose the kings in brackets:

<font color = "red">
  
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830; [K&#9830;]</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824; [K&#9824;]  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827; [K&#9827;]

That's 3 out of 39 or 3/39 which reduces to 1/13

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Find the probability of a nine given the card is a face card.

Start with a full deck:
<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824;  K&#9824;  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827;  K&#9827; 

Remove all cards except the ones that are given.
We are given that it is a face card, so we remove everything but the
face cards.  So we have only these 12:

<font color = "red">
                               J&#9829;  Q&#9829;  K&#9829; 
                               J&#9830;  Q&#9830;  K&#9830;</font>
                               J&#9824;  Q&#9824;  K&#9824;  
                               J&#9827;  Q&#9827;  K&#9827; 

I'll enclose the nines in brackets.  Whoops!  None of them are nines!

That's 0 out of 12 or 0/12 which reduces to 0.

That one was obviously zero because if you're given it's a face card,
then it is impossible that it is a 9, because a 9 is not a face card.

Edwin</pre>