1. Random Numbers of Trials

As in lecture and the textbook, we will use " -coin" to mean a coin that lands heads with probability .

a) In a randomized controlled experiment, it costs researchersdollars to study each subject in the treatment group anddollars to study each subject in the control group. Suppose each ofsubjects is randomized into one of the two groups according to whether a -coin lands heads (treatment) or tails (control). Find the expected total cost to study allsubjects.

b) I toss -coins. Those that land tails I toss again, and then I stop. Find the expected total number of heads.

c) I have one 0.25-coin, one fair coin, and three 0.75-coins. I pick one of the five coins at random and toss it till I get 10 heads. Find the numerical value of the expected number of tosses.