From a group of 20 people (including Alice and Bob), 5 are randomly
selected. Assuming that Alice was selected, what is the probability
that Bob will not be selected?
This is the question I faced in my probability course.
My solution was to calculate all possible selections when Alice is selected: ${19 \choose 4}$, when both Alice and Bob are selected ${18 \choose 3}$.
Then, I calculated the probability for Bob not to be selected if Alice is selected:
$$1-{\frac{18 \choose 3}{19 \choose 4}} = {\frac{15}{19}} \approx 0.78947$$
I would be happy if someone will confirm that this solution is correct, or tell me what is the correct solution.
Best Answer
Your result is correct.
Another way is to say that, if Alice is selected, there are $4$ places left and $19$ people so $15$ will not be selected, and each of the $19$ is equally likely to not be selected, making the probability of Bob not being selected $\dfrac{15}{19}$