# Is our teacher speaking correct about the probability of a national lottery ticket?

I'm trying to keep the question short. Today the teacher said something like this.Your chance of winning in the national lottery does not depend on which city you bought the ticket from. If you want, buy your ticket from city "A" or buy your ticket from city "B". Your probability of winning does not change.A criticism was made to the teacher: But the people who win are always those who live in city "A". The teacher replied as follows. Because the number of tickets sold in city "A" is more than the number of tickets sold in city "B". However, no matter which city you buy a ticket from, your chances of winning will always remain the same.After some thought, I made an objection.

I said, this argument is valid only and only if:

If the number of tickets DISTRIBUTED to be sold to city "A" and "B" is the same, then my chances of winning will not change. Then, the probability of winning for people living in city "A" may be attributed to the over-selling of tickets. But even this argument is not valid if the winning ticket should only have $$1.$$ Because the number of distribution and purchase of the ticket to cities is not the same thing.

My question: Is the teacher right? What am I getting wrong? Where am I making a mistake?

Thank you!

• It depends on the lottery. One sort of lottery you choose the numbers and to win they must match the specific numbers drawn. So unless people in City A or B are more likely to choose a particular number the city does not matter. Dec 4 '20 at 5:59
• If one city has exactly one inhabitant and the other has 10 million then it is much more likely that a winner will be found in the big city, right? Dec 4 '20 at 6:00
• @copper.hat thank you for explaning. But city A has 1000 tickets, city B has 10 tickets. If I buy tickets from city A or city B, the probability is the same? (we cannot write the numbers of lottery tickets ourselves)
– user548054
Dec 4 '20 at 6:07
• @copper.hat Thank you. You explained it very well without a formula. Now I think I can understand.
– user548054
Dec 4 '20 at 6:23
• Good luck. See Dhanvi's answer for a formula based answer. Dec 4 '20 at 6:24

I think the confusion between conditional and absolute probability - the chance that a particular lottery number is the winning number is the same regardless of which city you purchase it from, and I will demonstrate it

Suppose there are $$N$$ total numbers, out of which only one is the winning ticket. Let $$N_A$$ and $$N_B$$ be the number of tickets that go to cities $$A$$ and $$B$$ respectively

Hence Probability that winning ticket goes to city A is $$P_A = \frac{N_A}{N}$$, and similarly for city B is $$P_B = \frac{N_B}{N}$$

Now, the probability that someone wins it from city A is

$$\text{Probability winner from City A} = \text{Probability of drawing winning ticket from city A} \times \text{Probability winning ticket is in City A}$$

$$\implies P(A) = \frac{1}{N_A} \times \frac{N_A}{N} = \frac{1}{N}$$

So the effect of "big city" vs "small city" does not really matter

• (+1) Thank you .
– user548054
Dec 4 '20 at 6:27