In my country, playing the lottery during the festive season is very common. I am trying to find the probability of winning the second prize and third prize in some lottery draw.
There are $49$ balls that can be selected. A person buys a ticket by buying a ticket with $7$ numbers on it. Of which, $1$ number is selected as the "Special" number.
A draw is done when $6$ "Normal" balls are drawn, with $1$ ball drawn as the "Special" ball.
Second prize in won when a ticket's "Special" number matches and any $5$ of $6$ "Normal" balls match.
Third prize is won when a ticket's "Special" number matches and any $4$ of $6$ "Normal" balls match.
What is the probability of winning second prize or third prize? I know it is not as simply as $\frac{6!}{49!}$.
Best Answer
There are $49{48 \choose 6}$ different draws. You can win second prize in $6\cdot 42=252$ ways-choose one of the non-special numbers to miss and $42$ ways to pick the undrawn number. You can win third prize in ${6 \choose 2}{42 \choose 2}=12915$ ways. So the chances are second prize about $4.2\cdot 10^{-7},$ third prize about $2\cdot 10^{-5}$