lesson 9.11: why is it written as 'eine', not 'einer'(dative feminine, my logic)

Hello: lesson 9.11: why is it written as 'eine', not 'einer'(dative feminine, my logic). Thank you. 
Ich gebe Paul eine Fahrkarte für den Bus.

I give Paul a ticket for the bus.

Hallo AhmetO3,

eine Fahrkarte is the direct object of the sentence and therefore has to be put the accusative case. 
Paul is the person to whom an action is being performed and so he is acting as the indirect object of the sentence and has to be put in the dative case. 
If you wanted to put a definite article in front of  his name it would have to be declined to dem: "Ich gebe (dem) Paul eine Fahrkarte für den Bus."  - "I give (the) Paul a ticket for the bus." Please note that this would be quite uncommon to say.

Let me give you some more examples:
"Ich gebe dem Mann eine Fahrkarte für den Bus." - "I give the man a ticket for the bus."
"Ich gebe der Frau eine Fahrkarte für den Bus." - "I give the woman a ticket for the bus."
"Ich gebe dem Kind eine Fahrkarte für den Bus." - "I give the child a ticket for the bus."

I hope this clears things up!

Grüße
Julia

As an addition  to Julia's comprehensive reply, you could apply a "rule of thumb": if you can insert "to" or "for" in front of the noun it is likely to be the indirect object.
In the above:
 "I give to the man a ticket for the bus." (The order is a bit unusual and sounds stilted, more common would be "I give a ticket to the man for the bus.)
And in Ahmet's original, you could say "I give a ticket to Paul for the bus."
In the last example it's clearer that "ticket" is the direct object, I think.