Discrete Random Variables (DRV) - Binomial Probabilities Using Technology

Pauline frequently uses a social networking website to chat with her friends in the evening. On this site, she has a total of \(120\) friends. If she logs on to the site at \(9.30\) pm, on average, \(35\)% of her friends will also be logged on to the site.

Pauline logs on to the site at \(9.30\) pm on a particular day. The number of her friends who are also logged on to the site is denoted by \(L\).

1. State, in context, two assumptions needed for \(L\) to be well modelled by a binomial distribution.

2. Explain why one of the assumptions stated in part (i) may not be valid in this context.

Assume now that these assumptions do in fact hold.

3. If \(8\) of her friends on this site are chosen at random, find the probability that more than half of them are logged on to the site.