Answer
$\frac{14875}{12550824}$
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 2 numbers out of 5, or $_{5}C_2$, leaving 3 balls in $51$ (non-winning balls) or $_{51}C_3$. Thus the total number of ways for this part is $_{51}C_3\times_{5}C_2=208250$
Step 5. There is only one way to select the gold ball; thus the total number of ways of winning is $208250$
Step 6. The probability of winning this prize is
$p=\frac{208250}{46\times_{56}C_5}=\frac{208250}{46(3819816)}=\frac{14875}{12550824}$
