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MODELING WITH MATHEMATICS A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for I point with a kick (which is successful 99\(\%\) of the time) or 2 points with a run or pass (which is successful 45\(\%\) of the time). a. If the team goes for 1 point after each touchdown, what is the probability that the team wins? loses? ties? b. If the team goes for 2 points after each touchdown, what is the probability that the team wins? loses? ties? c. Can you develop a strategy so that the coach's team has a probability of winning the game that is greater than the probability of losing? If so, explain your strategy and calculate the probabilities of winning and losing the game.

Solve the equation. Check your solution. $$0.3 x-\frac{3}{5} x+1.6=1.555$$

Tell whether the question can be answered using permutations or combinations. Explain your reasoning. Then answer the question. To complete an exam, you must answer 8 questions from a list of 10 questions. In how many ways can you complete the exam?

Compare the quantities \(_{50} C_{9}\) and \(_{50}C_{41}\) without performing any calculations. Explain your reasoning.

You work 5 evenings each week at a bookstore. Your supervisor assigns you 5 evenings at random from the 7 possibilities. What is the probability that your schedule does not include working on the weekend?

Find the probability of winning a lottery using the given rules. Assume that lottery numbers are selected at random. You must correctly select 4 numbers, each an integer from 0 to 9 . The order is important.

A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Consider a convex polygon with \(n\) sides. a. Use the combinations formula to write an expression for the number of diagonals in an \(n\) -sided polygon. b. Use your result from part (a) to write a formula for the number of diagonals of an \(n\) -sided convex polygon.

Every student in your history class is required to present a project in front of the class. Each day, 4 students make their presentations in an order chosen at random by the teacher. You make your presentation on the first day. a. What is the probability that you are chosen to be the first or second presenter on the first day? b. What is the probability that you are chosen to be the second or third presenter on the first day?Compare your answer with that in part (a).