A lottery ticket consists of two rows, each containing $3$ numbers from $1,2,........,50$. The drawing consists of choosing $5$ different numbers from $1,2,......50$ at random. A ticket wins if its first row contains at least two of the numbers drawn and its second row contains at least two of the numbers drawn. The four examples below represent the four types of tickets:
| Ticket 1 |
Ticket 2 |
Ticket 3 |
Ticket 4 |
| 1 2 3 |
1 2 3 |
1 2 3 |
1 2 3 |
| 4 5 6 |
1 2 3 |
2 3 4 |
3 4 5 |
for example, if the numbers $1,3,5,6,17$ are drawn, then Ticket 1 , Ticket 2, and Ticket 4 all win, while Ticket 3 loses. Compute the winning probabilities for each of the four tickets.
