Answer
$\frac{2125}{29285256}$
Work Step by Step
Step 1. Choose 5 numbers from $1\ to\ 56$. We have $_{56}C_5$ combinations.
Step 2. Choose 1 number from $1\ to\ 46$. We have 46 choices.
Step 3. The total sample space is $46\times_{56}C_5$
Step 4. To win the prize, we need to match 3 numbers out of 5 or $_{5}C_3$, leaving 2 balls in $51$ (non-winning balls) or $_{51}C_2$. Thus the total ways for this part is
$_{51}C_2\times_{5}C_3=12750$
Step 5. There is only one way to select the gold ball. Thus the total ways of winning is $12750$
Step 6. The probability of winning this prize is
$p=\frac{12750}{46\times_{56}C_5}=\frac{12750}{46(3819816)}=\frac{2125}{29285256}$
