The follow question sets up a scenario for a random variable:
The question is in a section on the binomial probability distribution. However, I don't believe that this random variable can be modelled as having a binomial distribution of probabilities. This is because if you are randomly selecting members of a group, then the the composition of the existing sample affects the composition of the remaining, non-sampled members of the group. Therefore, the probabilities are not independent of each other, and there is not a fixed probability of success. This means the binomial distribution cannot be used.
Unless, that is, the sample gets replaced, which would mean new people join the book club for every person sampled. Or unless people can be part of the same sample multiple times. But, I don't think either of those would make sense.
I believe the context given in this question resembles this: Combinatorial Probability vs Binomial Probability
However, this is the answer to a) given in the textbook:
Have they made a mistake, or are they asking you to ignore the realities of the context and accept that there is this supposed fixed probability of success?
The population probability for being left-handed is $p=0.15$ and this is fixed and constant. You then have a sample of that population and that sample contains anywhere from $X=0$ to $X=20$ left-handers. With the assumptions given, a binomial model for $X$ is suitable.
Maybe you are thinking of the context of a pre-filled 'bag' containing $a$ right-handed and $b$ left-handed people, where the probabilities for sampling left- or right- handed people from the bag change as you sample/remove people from the bag.