My son is planning a party for his upcoming 21st birthday. Under COVID restrictions, the venue he has chosen has a limit of 30 guests per function, meaning he can have no more than 29 friends attend. Each of his friends has a chance of being unavailable on the day. Suppose he has 38 friends he wants to invite, each with a 22% chance of having a conflicting engagement and thus not able to attend.

1. (a)Using a normal approximation without continuity correction, estimate the chance that, if he invites all 38 friends, the number of friends that can attend will exceed the allowable limit. Recalculate your estimate using the continuity correction.
2. (b)Obviously, he would like to minimise the chance of having too many guests for the venue; however, if there are less than 20 attendees (i.e., less than 19 of his friends can make it), the venue will add a surcharge (to discourage unused capacity as it diminishes overall profitability). Using a normal approximation with continuity correction, determine the number of invitations my son should send out to minimise the total chance of either too many or too few guests. [NOTE: Excel will be a more useful tool for this problem than calculus.]