SOLUTION: You are playing go fish with a friend and you have the cards 7,7,5,5,2,an Ace and a King your friend has seven cards too and you are playing with a set of fifty two cards what is t

We define the sample space which is the set of combinations of 7 cards
from the 52-7 or 45 cards besides the 7 cards in your hand.  

So the denominator of your desired probability is 45C7 = 45379620

We will first find the probability of the complement event and then 
subtract its probability from 1.

The complement event is:
"Your friend's combination of 7 cards does not contain a seven, or a five, or
a two, or an Ace, or a King. 
 
Now among the 45 cards not in your hand, there are 2 sevens, 2 fives, 3 twos,
3 Aces, and 3 Kings. That's 2+2+3+3+3 = 13 cards that match one of your cards,
and therefore 45-13 or 32 which don't match any of your cards.
 
So in order for your friend not to have a seven or a five, or a two, or an Ace,
or a King, your friend must have a combination of 7 cards from those remaining
32 cards.

So there are 32C7 = 3365856 possible 7-card hands he or she could have from the
32 cards that do not match any of the cards in your hand.

So the numerator of the probability of the complement event is



So the desired probability is 1 - .0741710927 = .9258289073.

So there's a 92.6% chance that your friend has some card of the same rank as
one of yours.  That's pretty likely.

Edwin