Five tickets are bought in a raffle containing eighty sold. There are three prizes. Tim buys five. What is the chance Tim wins exactly one prize? I've got the solution and answer manual, but don't want to spend all that time drawing tree diagrams etc.
[Math] The chance of winning exactly one prize after buying 5 tickets out of 80, with 3 tickets winning
probability
Best Answer
The probability that the first prize is won is $5/80$. Now, the probability that only the first prize is won (that is, the first prize is won, and at the same time the second and third are lost) is $$ \frac{5}{80}\cdot \frac{75}{79}\cdot \frac{74}{78} = 0.0563 = 5.63\% $$ The probability that only the second prize is won is $$ \frac{75}{80}\cdot \frac{5}{79}\cdot \frac{74}{78} $$ which you can see has exactly the same result. And finally, the probability of winning only the third prize is $$ \frac{75}{80}\cdot \frac{74}{79}\cdot \frac{5}{78} $$ which again has the exact same answer. All together, the probability that one of these three events occur (seing as how they cannot occur together, and are therefore disjoint) is just $$ 3\cdot 5.63\% = 16.89\% $$