Contestants on the television quiz show Who Wants to be a Millionaire can win $1 million if they answer 15 questions correctly. Each question is multiple choice with four possible answers. What are the odds of winning the million by guessing the answers to all 15 questions?

The producers of the show want to make it more likely to succeed. They make the first five questions much easier than the later questions. Also, each contestant has three lifelines that can often help the contestant to answer correctly. Suppose the contestant gets the first five
questions correct and of the remaining ten questions, three are answered correctly using lifelines. Of the remaining questions, the contestant knows the answer to three and for each of the others can identify two of the possible answers as being incorrect. What are the odds of winning the million dollars by guessing these two-part questions?

There are 15 questions each with odds of 1/4 for guessing correctly. The
odds of guessing all 15 correctly are (1/4)'5. The odds can be estimated without using a calculator simply by noting:
415 = (22)15 = 230 = (210)3
21° is very close to 1000 (2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024). Thus the odds are approximately 1 chance in 1000 * 1000 * 1000
or about 1 chance in a billion.
The denominator 415 can be evaluated more precisely using a calculator. Enter 4 and press the 37' key. Enter 15 and press the yX key again. The answer is 1,073,741,824.
Thus, there is one chance in 1,073,741,824.
The odds for the second question are much better.
Start with 15 questions.
Answer the first 5 (easy questions) correctly.
Answer 3 of the remaining 10 questions correctly using lifelines. Know the answers to 3 of the remaining 7 questions.
Eliminate 2 out of 4 answers for each of the remaining 4 questions. That leaves 4 two-part questions.
The odds for guessing these 4 correctly are:
(1/2) * (1/2) * (1/2) * (1/2) = (1/2)4
This evaluates to one chance in 16. (Much, much better that 1 in a billion.)