Question 697405
A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is 

(i)a card of spade or an ace 
<pre>
This asks for the probability of drawing one of these 16
<font color = "red">
 
A&#9829;   
A&#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;   

out of these 52:

<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; 

In probability theory "out of" means over, so "16 out of 52" means
"16 over 52" and that is {{{16/52}}} and that reduces to {{{4/13}}}
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</pre>
(ii)a red king 
<pre>
This asks for the probability of drawing one of these 2
<font color = "red">
                                                        K&#9829; 
                                                        K&#9830;</font>
out of these 52:

<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; 

In probability theory "out of" means over, so "2 out of 52" means
"2 over 52" and that is {{{2/52}}} and that reduces to {{{1/26}}}
------------------------------------------------------------------
</pre>
(iii)neither a king nor a queen
<pre>
This asks for the probability of drawing one of these 44
<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;   
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  </font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;    
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;   

out of these 52:

<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; 

In probability theory "out of" means over, so "44 out of 52" means
"44 over 52" and that is {{{44/52}}} and that reduces to {{{11/13}}} 
--------------------------------------------------------------------
</pre>
(iv)either a king or queen.

<pre>
This asks for the probability of drawing one of these 8
<font color = "red">
 
                                                    Q&#9829;  K&#9829; 
                                                    Q&#9830;  K&#9830;</font>
                                                    Q&#9824;  K&#9824;  
                                                    Q&#9827;  K&#9827;    

out of these 52:

<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;

In probability theory "out of" means over, so "8 out of 52" means
"2 over 52" and that is {{{8/52}}} and that reduces to {{{2/13}}}
------------------------------------------------------------------ 
Notice that (iii) and (iv) are complement events, so we could have
gotten the answer to (iv) by subtracting the probability of (iii)
from 1:

1 - {{{11/13}}} = {{{13/13}}} - {{{11/13}} = {{{2/13}}}.

It would have been easier to have done (iv) first and then gotten the
answer to (iii) by

1 - {{{2/13}}} = {{{13/13}}} - {{{2/13}} = {{{11/13}}}.

Edwin</pre>