# how to calculate probability of winning exactly one number at lottery

There are 38 numbers, I pick 7. What are probabilities I win exactly one number, no more and no less.

Is this possible to calculate.

It would be simple to calculate what are the chances I don't win any number, but how can you calculate chances of winning exactly one?

• To be clear, are you saying that you choose 7 numbers between 1 and 38, and there are 7 winning numbers, and there is exactly one match? If so, are repetitions allowed i.e. can you choose (1,1,1,1,1,1,1)? – pwerth Jan 16 at 19:30
• @pwerth: yes, 7 numbers between 1 and 38 and no repetions are allowed. If 16 is selected, it's eliminated. – Stenga Jan 16 at 19:32

You have chosen $$7$$ numbers. In order for you to match exactly one number, the winning numbers must include $$6$$ of the $$31$$ numbers you did not choose. $$\frac{7\cdot\binom{31}{6}}{\binom{38}{7}}\approx40.839\%$$