# % increase in probability of winning lottery by buying 1 extra ticket

I would like to know and be interested to see how you calculate:

The probability of winning a lottery with 5 numbers in any order between 1 a 50 AND two numbes betwen 1 - 11

The probability of winning a lottery with 5 numbers in any order between 1 and 49 AND 1 number between 1 and 10.

THEN: the impact on probability by buying 1 or more extra tickets

-
I would like to know how you calculate these numbers. How far can you get? Where do you get stuck? –  Gerry Myerson Feb 19 '12 at 7:42
Since the probability of winning a lottery jackpot by buying one ticket is so low, the probability of winning with a small number $n$ of tickets will be approximately $n$ times as high, providing you do not deliberately have the same set of numbers. –  Henry Feb 19 '12 at 9:35

Lets denote total number of numbers as : $n$ , and number of numbers that can be drawn as $k$ .

If you buy only one ticket you can calculate probability using following formulae :

$$P_1= \left(\binom {n}{k}\right)^{-1}$$

$$\text{where} : \binom {n}{k} = \frac{n!}{(n-k)!\cdot k!}$$

If you buy two tickets you can calculate probability using formulae :

$$P_2=2P_1=2 \cdot \left(\binom {n}{k}\right)^{-1}$$

If you buy $~m~$ tickets you can calculate probability using formulae :

$$P_m=mP_1=m \cdot \left(\binom {n}{k}\right)^{-1}$$

-
This ignores the extra numbers on the ticket, and the second half is an approximation –  Henry Feb 19 '12 at 9:36