The FLAMES-ARE-US Co. manufactures butane cigarette lighters. Your top-of-the-line lighter, which has the brand name "SURE-FLAME," costs \(\$ 29.95\). As a promotional strategy, the SURE-FLAME lighter carries a guarantee that if it takes more than five attempts before the lighter actually lights, then the customer will be given \(\$ 1,000,000\). The terms of the guarantee require that the demonstration of failure of a SURE-FLAME lighter to light within five attempts must be witnessed by an official of the company, and each original buyer of a new SURE-FLAME lighter is allowed only one attempt at being awarded the \(\$ 1,000,000\). The lighter is such that each attempt at lighting the lighter has a probability of success (it lights) equal to .95, and the outcomes of attempts to light the lighter are independent of one another.

(a) Define the appropriate probability space for the experiment of observing the number of attempts necessary to obtain the first light with the SURE-FLAME lighter.

(b) What is the probability that a buyer who attempts to demonstrate the failure of the SURE-FLAME to light in five attempts will actually be awarded \(\$ 1,000,000\) ?

(c) What is the expected number of attempts required to obtain a light with the SURE-FLAME lighter?

(d) What is the expected value of the award paid to any consumer who attempts to claim the \(\$ 1,000,000\) ?