A local outfitter has thirty-two kayaks for weekend rental. On any given weekend, outfitter takes $40$ reservations for the $32$ kayaks, as he experiences "no-shows" - those people with reservations who do not pick up the kayak. He estimates the probability of any reservation being a "no-show" is $0.30$. As a result, the outfitter faces the possibility of "overbooking". Any person making a reservation who finds himself or herself without a kayak is provided with a free rental the following weekend - a policy that the outfitter clearly states when a reservation is made. (There are no guarantees, so pick up your reserved kayak early!)
a) During a fully booked weekend what is the probability that every person showing up to pick up his or her reserved kayak will have a kayak for the weekend?
b) During a fully booked weekend what is the probability that the outfitter overbooks?
For a), The logic I was using is that if he has a fully booked weekend, then there is 8 no shows. so $40\choose8$ $*$ $0.3^8$ * $0.7^{32}$, but this is not correct. And for b) I'm not sure how to approach it.
You did not account for the fact that everyone that shows up will have a kayak if there are $8$ or more no-shows.
a) Everyone will have a kayak for the weekend if the number of people that actually show up is $0,1,2,...,32$. Thus we have
$$P(\text{not overbooking})=\sum_{k=0}^{32} {40 \choose k}\left(0.7\right)^k \left(0.3\right)^{40-k}\approx0.945$$
This can be calculated using R statistical software:
b) Notice that this is just the compliment of question (a). The outfitter overbooks if the amount of people that actually show up is $33,34,...,40$. Thus we have
$$P(\text{overbooking})=\sum_{k=33}^{40} {40 \choose k}\left(0.7\right)^k \left(0.3\right)^{40-k}\approx0.055$$
This can be calculated using R statistical software: