1. (6 pts in total) There are n balls in a jar, labeled with the numbers 1, 2, .. ., n. A total of balls are drawn, one by one with replacement, to obtain a sequence of numbers. What is the probability that the sequence obtained is strictly increasing? 1) Sample space 1.1) (0.5 pts) Please give a possible final outcome of the process. 1.2) (1.5 pts) Design an experiment step-by-step to count the sample space (follow the story in the problem), and provide the number of possible outcomes for each step Step 1: # of options: Step 2: # of options: (Feel free to add more steps if needed) 1.3) (0.5 pts) Count the number of final outcomes in the sample space.2)Event 2.1) (0.5 pts) Define your event. If necessary, you might also want to define the complement of the event or a list of simpler events. Page 3 Name (print): 2.2) (2 pts) Design an experiment step-by-step to realize the event (to make sure the event occurs. you have to assume you have some super power, and don't need to follow the rule in the story). and provide the number of possible outcomes for each step. Hint: to make sure you have a strictly increasing sequence, you have to draw k different numbers and then sort this k numbers. Step 1: # of options: Step 2: # of options: (Feel free to add more steps if needed)2.3) (0.5 pts) Count the number of possible outcomes in your experiment. 3) (0.5 pts) Compute the probability of the event. Page 4 2. (9 pts in total) A bag contains one marble (marble 1) which is either green or blue, with equal probabilities. A green marble (marble 2) is put in the bag (so there are 2 marbles now), and then a random marble is taken out. The marble taken out is green. What is the probability that the two marbles are green?1) Define the three basic events in this story. Hint: the basic events are independent. A: marble 1 is green B: marble 1 is taken out C: marble 2 is green 2) (1.5 pts) What is P(A), P(B) and P(C)? 3) (1 pt) Express the event "a random marble is taken out and it is green' in the form of A, B, and C. Hint: the event "a random marble is taken out and it is green' is the same to say that "Marble 1 is green and Marble 1 is taken out' or 'Marble 2 is green and marble 2 is taken out'. 4) (1 pt) Express the event "two marbles are green' using the basic events. 5) (1 pt) What probability does the story look for? 6) (2 pts) Express the probability in 5) in the form of the probabilities of intersection(s) using thedefinition of conditional probability. Hint: axiom 2 or property 3 will be used here. Set operations can be found in the last page of the cheat sheet. 7) (2.5 pts) Compute the probability in 5) based on the results in 6). Hint: event A. B, and C are independent. P(An B n C) = P(An B'n C) = P(An B) =P(B' n C) = Your conditional probability in 5) is 3. (4 pts in total) An airline overbooks a flight, selling more tickets for the flight than there are seats on the plane (figuring that it's likely that some people won't show up). The plane has 100 seats, and 110 people have booked the flight. Each person will show up for the flight with probability 0.9, independently. Find the probability that there will be enough seats for everyone who shows up for the flight. 1) The story can be seen as a sequence of Bernoulli trials. You will need to figure out the basic elements in the Bernoulli trials. They are 1.1) The Bernoulli trial in this story: whether a person will show up 1.2) (0.5 pts) How do you define the success in one Bernoulli trial?1.3) (0.5 pts) What is the success rate? 1.4) (0.5 pts) How many trials are there? 1.5) (1 pts) Define the r.v. X: the number of persons showing up. What is the distribution of X? Use the name and the parameters to express the distribution. 2) (0.5 pts) User.v. X to express the event 'enough seats for everyone who shows up for the flight". 3) (1 pts) Compute P(X _ 100). It is OK to have binomial functions, geometric functions in the final answer.4. (4 pts in total) There are n people in a room. Assume each person's birthday is equally likely to b e any of the 365 days of the year, and that people's birthdays are independent. what is the expect ed number of distinct birthdays among the n people, i.e., the expected number of days on which a t least one of the people was born? 1) Define r.v. X: the number of days on which at least one of the people was born. 2) Define indicator .v.s /; = if at least one person is born on day) for j=1, ..., 365 otherwise 3) (2 pts) Please compute the expected value of Ij. 4) (1 pts) Please express X in the form of I/'s 5)(1 pts) Please compute E(X)