Consider a probability space whose sample space is the natural numbers

(i.e., 1,2,3, . . .). Show that not all the natural numbers can have equal probability.

11. Prove that any event is independent of the universal event and of the null

event .

12. Suppose is an elementary outcome, i.e., . What is the difference between and {}?. How many elements does have?. How many elements

does {} have?

13. You are a contestant on a television game show. Before you are three closed

doors. One of them hides a car, which you want to win; the other two hide

goats (which you do not want to win).

First you pick a door. The door you pick does not get opened immediately.

Instead, the host opens one of the other doors to reveal a goat. He will then

give you a chance to change your mind: you can switch and pick the other

closed door instead, or stay with your original choice. To make things more

concrete without losing generality concentrate on the following situation

(a) You have chosen the first door.

(b) The host opens the third door, showing a goat.

If you dont switch doors, what is the probability of wining the car? If you

switch doors, what is the probability of wining the car? Should you switch

doors?

14. Linda is 31 years old, single, outspoken and very bright. She majored in

philosophy. As a student she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable?

(a) Linda is a bank teller?

(b) Linda is a bank teller who is active in the feminist movement

Name: 1. (15 points) Calculations under common probability distributions. Include one or two calculation steps such that you may still get the points even if the final numbers are wrong. (a) (5 points) Suppose that we have a biased coin with the probability of heads being 0.6. What is the odds ratio for heads? (b) (5 points) Suppose that the number of cars passing by an intersection is 3 every 10 minutes. What is the probability of observing no more than 2 cars in ten minutes? (c) (5 points) Suppose that the highest daily temperature in September follows a normal distribution with mean 70 and standard deviation 5. What is the 95 quantile (i.c. 95th percentile - the value with an upper-tail probability of 0.05)? 2. (15 points) (a) (10 points) Fill in the following table with type I error and type II error. Decision Do not reject Ho Reject No Ho true Ha trueD Question 1 4 pts (GenEd5) What is the difference between population parameters and sample statistics? Parameters are calculations related to the entire collection of elements, while statistics are calculations related to some sub-collection of those elements. Population parameters are constants; sample statistics are variables. Population parameters are naturally larger. Population parameters are variables; sample statistics are constants.A college is studying the probability students will take one or more statistics classes. The following probability distribution shows the likelihood students will take 1, 2, 3, 4, or 5 statistics classes. NOTE: This is NOT a Binomial Distribution. What is the mean number of statistics classes students take? Number of Statistics Classes Probability 1 60 2 .20 3 12 1 04 5 .04 0 2.50 1.72 Correct Answer Not Provided 3.003 (20 p) 45% of the students failed in statistics, 35% in computer and 25% in both statistics and computer courses. A randomly selected student, a) If it fails from the computer, the probability of failing the statistics, b) If it fails from statistics, the probability of failing from the computer, c) Find the probability of failing at least one of these two courses