Probability of winning at least one prize in a raffle

Question

"A guy bought $4$ tickets in a raffle of $200$ tickets, where $5$ of them gives a prize. What the probability of the guy winning at least $1$ prize".

My try

After a long analysis, i think the formula for exactly $1$ prize is:

$$\frac{\dbinom{5}{1}\dbinom{195}{3}}{\dbinom{200}{4}}$$

But the question is about at least $1$ prize, and i don't know if that affects my answer.

Is my answer right for exactly $1$ prize? How do i compute the answer for at least 1 prize?

Thanks.

Answer 1

When solving a question asking for the probability of at least 1 event, it is a good rule of thumb to first solve the complement -- the probability of 0 events occurring -- and then subtract from 1.

In this case, we want to find the probability that, given a guy has bought 4 raffle tickets, he wins 0 prizes.

If we logically walk through this, this is equal to the event that the all the 5 prizes go to a raffle ticket other than the 4 the guy has.

The probability of a single prize going to another raffle is $\frac{196}{200}$. This implies the probability of all 5 prizes going to another raffle is $(\frac{196}{200})^5$.

So the probability of winning at least 1 prize is equal to $1 - (\frac{196}{200})^5$.

Edit

In the previous solution, I made the erroneous assumption that this was a raffle drawn with replacement: that is, once a ticket has been selected, it is placed back into the bin.

To solve for a raffle drawn without replacement, we have to make a slight correction to the probabilities.

The probability of the event of the first prize going away is still $\frac{196}{200}$. However, as the winning ticket is thrown away, the probability of the event of the second prize going away is now $\frac{195}{199}$. And so forth.

In total, the probability of all 5 prizes going to another raffle will be $$\frac{196 \cdot 195 \cdot 194 \cdot 193 \cdot 192}{200 \cdot 199 \cdot 198 \cdot 197 \cdot 196}.$$

The probability, then, of winning at least 1 prize is the complement: $$1 - \frac{196 \cdot 195 \cdot 194 \cdot 193 \cdot 192}{200 \cdot 199 \cdot 198 \cdot 197 \cdot 196}.$$