First, we need to determine how much money Joseph has left after he buys the hat.
He has $34.25 and the hat costs $12.50.
Subtract the cost of the hat from his total money:
\[ 34.25 - 12.50 = 21.75 \]
Now, Joseph has $21.75 left to spend on tickets. Each ticket costs $7.25. Now we divide the remaining amount by the price of one ticket to find out how many tickets he can buy:
\[ \frac{21.75}{7.25} \approx 3 \]
Joseph can buy 3 tickets.
So, the answer is: 3 tickets.