The Teatown lottery has $30$ balls, numbered by the integers from $1$ to $30$. The lottery works as follows. Each contestant picks $4$ numbers of their choice from this range. In particular, Alice chooses the numbers $1, 2, 3$ and $5$, whilst Bob chooses the numbers $2,3, 5$ and $28$. The lottery operator then draws $5$ balls from the bag, without replacement. Give, with justification, exact expressions for the following probabilities.
$(i)$ The probability that the balls drawn are $2, 3, 5, 7$ and $11$.
$(ii)$ The probability that Alice wins the jackpot by having all of her chosen numbers drawn.
$(iii)$ The probability that exactly $2$ of Alice’s chosen numbers are drawn.
$(iv)$ The probability that Alice and Bob both have exactly $3$ of their numbers drawn.
Can you please check all my answers please ? Normally I would find the question okay but I am getting confused because they have told us exactly what numbers Alice and Bob have drawn already.
$(i)$ The probability that the balls drawn are $2, 3, 5, 7$ and $11$.
In total, there are $30C5$ possibilities of selecting $5$ balls from the bag unordered without repetition. The numbers $2, 3, 5, 7$ and $11$ form one of these combinations, so probability is $\frac{1}{30C5}$.
$(ii)$ The probability that Alice wins the jackpot by having all of her chosen numbers drawn.
So for the $5$ balls drawn from the bag, $4$ of them must be $1,2,3,5$. For the $5$th ball, there are $26$ possibilities from the remaining balls. So total probability, using working from $(i)$, is $\frac{26}{30C5}$.
$(iii)$ The probability that exactly $2$ of Alice’s chosen numbers are drawn. From the $4$ balls drawn from Alice, $4C2$ are also drawn from the bag. Then for the remaining $3$ balls drawn from the bag, there are $26C3$ possibilities. Probability is $\frac{4C2 \cdot 26C3}{30C5}$.
$(iv)$ The probability that Alice and Bob both have exactly $3$ of their numbers drawn.
I know this is the hardest part so as long as I am sure on how to do the first $3$ I can focus on this part myself after. I know I need to consider cases, like when the correct balls are $2,3,5$ since these are the $3$ balls Alice and Bob have in common. I could maybe use inclusion-exclusion.