# What's the probability of winning 10 lotteries with the same number?

Assuming I play a Lottery where I can buy a slip with exactly one random number from $$1$$ to $$10$$. The Probability of winning this lottery is $$\frac{1}{10}$$, as only $$1$$ winning number will be drawn.

Now I add a second lottery game on this Lottery Slip (again $$1$$ random number from $$1$$ to $$10$$). But, when I draw a winning number, this counts for both lotteries.

Example $$1$$: My Lottery Slip says "Game $$1$$: $$5$$ and Game $$2$$: $$8. \implies$$ Winning Number is $$8\implies$$ I won 2nd Lottery.

Example $$2$$: My Lottery Slip says Game $$1$$: $$8$$ and Game $$2$$: $$8. \implies$$ Winning Number is $$8$$ $$\implies$$ I won both Lotteries.

Example $$3$$: My Lottery Slip says Game $$1$$: $$1$$ and Game $$2$$: $$8. \implies$$ Winning Number is $$7$$ $$\implies$$ I lost both Lotteries.

What is the probability of at least winning once with a Lottery Slip of $$10$$ such games? Why?

• Use mathjax please, and edit, I have done some part. Sep 17, 2020 at 12:22
• Sry, I am new to this forum. So I am not very familiar with mathjax. Tried my best now to edit.
– Max
Sep 17, 2020 at 12:31
• Your question seems unclear , how do you define winning a game ? Winning both the draws ? @Max Sep 17, 2020 at 12:33
• The thing that confuses me is that, aren't winning numbers tied to the game? I mean, if you have two rounds of games, doesn't that mean that there would be two winning numbers? Sep 17, 2020 at 12:37
• Basically, isn't it one lottery and you buy 10 tickets (each with a random number)? Sep 17, 2020 at 12:38

However, you have ten of these "double lottery" games. That means the not winning rate is $$0.9\cdot 0.9 = 0.81$$, since you have two independent picks in one game. This occurs $$10$$ times, so $$0.81^{10}\approx 0.1215$$. Subtracting from $$1$$ gets the answer of $$1-0.1215 = 0.8785 = \boxed{87.9\%}.$$