Innocent until proven guilty? In Japanese criminal trials, about 95% of the defendants are found guilty. In the United States, about 60% of the defendants are found guilty in criminal trials. (Source: The Book of Risks, by Larry Laudan, John Wiley and Sons) Suppose you are a news reporter following twelve criminal trials. (For each answer, enter a number.)

(a) If the trials were in Japan, what is the probability that all the defendants would be found guilty? (Round your answer to three decimal places.)

What is this probability if the trials were in the United States? (Round your answer to three decimal places.)

(b) Of the twelve trials, what is the expected number of guilty verdicts in Japan? (Round your answer to two decimal places.) verdicts

What is the expected number in the United Sates? (Round your answer to two decimal places.) verdicts

What is the standard deviation in Japan? (Round your answer to two decimal places.) verdicts

What is the standard deviation in the United States? (Round your answer to two decimal places.) verdicts

*I am needing help inputting this equation(s) into Excel

3. You receive emails by a Poisson Arrival Process at a rate of 12 emails per hour.

(a) (6 points) Find the probability that you receive exactly 3 emails between 4:10 PM and 4:20 PM.

(b) (6 points) You start checking your email at 10:00 AM. What is the expected time of your first email?

(c) (9 points) Given that you receive exactly 10 emails between 4:00 PM and 5:00 PM, what is the (conditional) distribution of the number of emails you receive between 4:45 PM and 5:00 PM? For full credit, name the distribution and its parameters.

(d) (9 points) You read the emails you received between 10:00 AM and 11:00 AM and respond to them independently with probability 1/3.Let N be the number of emails you receive during that time window, and M be the number of emails you respond to. What is P(N= 0|M= 0)? (For full credit, your final answer should be in “closed form” and not include a summation.)

The owner of a local car dealership has just received a call from a regional distributor stating that a $5000 bonus will be awarded if the owner's dealership sells at least 10 new cars next Saturday. On an average Saturday, this dealership has 75 potential customers look at new cars, but there is no way to determine exactly how many customers will come this particular Saturday. The owner is fairly certain that the number would not be less than 40, but also thinks it would be unrealistic to expect more than 120 (which is the largest number of customers to ever show up in 1 day). The owner determined that, on average, about one out of ten customers who look at cars at the dealership actually purchase a car - or, a .10 probability (or 10% chance) exists that any given customer will buy a new car.

a. Create a spreadsheet model for the number of cars the dealership might sell next Saturday

b. What is the probability that the dealership will earn the $5000 bonus?

c. If you were this dealer, what is the maximum amount of money you would be willing to spend on sales incentives to try to earn this bonus?

A professional basketball player makes 88% of the free throws he tries. Assuming this percentage will hold true for future attempts, find the probability that in the next 8 tries, the number of free throws he will make is exactly 8.

Round your answer to four decimal places.

P (exactly 8)= ???

1. You roll two fair dice, and denote the number they show by X and Y. Let U = min{X, Y } and V = max{X, Y }. Write down the joint probability mass function of (U, V ) and compute ρ(U, V ) i.e the correlation coefficient of U and V

The mean number of words per minute (WPM) typed by a speed typist is 72 with a variance of 100. What is the probability that the sample mean would differ from the population mean by more than 3.3 WPM if 43 speed typists are randomly selected? Round your answer to four decimal places.

Naval intelligence reports that 8 enemy vessels in a fleet of 21 are carrying nuclear weapons. If 10 vessels are randomly targeted and destroyed, what is the probability that at least 7 vessels transporting nuclear weapons were destroyed? Express your answer as a fraction or a decimal number rounded to four decimal places.

Naval intelligence reports that 8 enemy vessels in a fleet of 21 are carrying nuclear weapons. If 10 vessels are randomly targeted and destroyed, what is the probability that at least 7 vessels transporting nuclear weapons were destroyed? Express your answer as a fraction or a decimal number rounded to four decimal places.

In 2003, it was reported that 40% of all car accidents in Connecticut were alcohol related. Suppose we take a random sample of 100 car accidents in Connecticut and let X be the number that are alcohol related.

Find the probability that at least 40 were alcohol related and at least 40 were not alcohol related.

Naval intelligence reports that 5 enemy vessels in a fleet of 17 are carrying nuclear weapons. If 7 vessels are randomly targeted and destroyed, what is the probability that no more than 1 vessel transporting nuclear weapons was destroyed? Express your answer as a fraction or a decimal number rounded to four decimal places.