Suppose you are given a test composed of ten problems. Not all problems on the test carry the same weight. The first eight problems are worth as much...

Suppose you are given a test composed of ten problems. Not all problems on the test carry the same weight. The first eight problems are worth as much as each other and in total sum to sixty points. The last two problems are also worth as much as each other and in total sum to forty points. Each problem is a TRUE/FALSE problem. In order to pass, you need to score at least sixty points on the test. You forgot to study for the test. You decide to guess the answer to each problem. Your strategy is to not answer false for any two consecutive questions. In how many ways can you answer all ten questions on the test?

2. Your friend Nathan also forgot to study and decides to just guess the answer to each problem. Given that Nathan is able to correctly guess the answer to the last two problems, what is the probability that he passes the test?

3. Find the probability that Nathan passes the test. [Hint: First condition on the number of correctly answered questions among the last two problems.]