There are N hostages and one bad guy, who needs a password from the hostages. One by one, the bad guy asks each hostage for the password. If the hostage squeals, the game ends; otherwise, the hostage dies, and the bad guy moves on to the next hostage.There are two types of hostages: 'sacrificial' types, and 'selfish' types. Sacrificial hostages receive payoff1 from dying, and9 if the bad guy gets the password. Selfish types get1 from dying, but don't care if the bad guy gets the password.Each hostage is sacrificial with probability 0.95, and selfish with probability 0.05.

(i) If the first hostage questioned is a sacrificial type, calculate this type's payoff from not giving up the password, in terms of N.

(ii) What is the maximum number of hostages required for the sacrificial type in part(i) to not give up the password? (Hint: logxn=nlogx)

(iii) Now assume that N= 50, and the probability each hostage is sacrificial is p. How large should p be to ensure that the sacrificial type does not give up the password?