To find the probability that a person who buys 2 raffle tickets will win the milking cow, we first need to determine the total number of raffle tickets sold.
Since the farming club will sell $6,000 worth of raffle tickets at $15 each, we calculate the total number of tickets sold as follows:
\[ \text{Total tickets sold} = \frac{\text{Total money from tickets}}{\text{Price per ticket}} = \frac{6000}{15} = 400 \]
Next, if a person buys 2 tickets, we need to find the probability that at least one of their tickets wins. The total number of favorable outcomes (winning tickets) when buying 2 tickets is 2 (since they have 2 tickets).
Now, the probability that a person does not win (all their tickets lose) can be calculated by using the total number of tickets:
- The total number of tickets is 400.
- The number of losing tickets is \(400 - 1 = 399\) (since there is only 1 winning ticket).
Now we calculate the probability that both tickets purchased by the person are losing tickets:
\[ P(\text{losing with 2 tickets}) = \frac{399}{400} \times \frac{398}{399} = \frac{398}{400} \]
Therefore, the probability that the person wins with at least one of their two tickets is:
\[ P(\text{winning with at least 1 ticket}) = 1 - P(\text{losing with both tickets}) = 1 - \frac{398}{400} = \frac{2}{400} = \frac{1}{200} \]
Thus, the probability that a person who buys 2 tickets will win the milking cow is:
\[ \boxed{\frac{1}{200}} \]