Sample Exam #1 Probability and Statistics I

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Edgar Lambert
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1 Sample Exam # Probability and Statistics I Instructions. Show all work. No need to simplify answers.. 0 points Jack and Jill, along with four other students chosen from the same class of ten students, form a line. How many lines can one make if Jill is always closer to the front of the line then Jack is? Solution: First chose four students from the remaining eight students in the class. There are 8 70 ways of doing this. Add to these students two 6! identical unisex mannequins. Then there are 60 ways of putting!!!!! the students and mannequins in a line. Next replace the the mannequin closest to the front of the line with Jill and the other mannequin with Jack. The answer is 7060, points Nine Reese s Peanut Butter Cups are to be distributed to six children. Two of the children are twins one pair of twins and each twin must be given the same number of Reese s as the other twin. How many ways can the Reese s be distributed it is possible to give some children no Reese s? Solution: Duct tape the two twins together and count them as one child now there are five children. We need to make sure the twin unit gets an even number of Reese s and the rest go to the four other children. # Reese s to twins # Reese s to other children # outcomes Total 0

Probability and Statistics I Sample Exam #. 0 points Let A, B and C be events such that A and B are independent, B and C are mutually exclusive, P A /, P B /6, and P C /. What is P A B C C?

2 Probability and Statistics I Sample Exam #. 0 points Let A, B and C be events such that A and B are independent, B and C are mutually exclusive, P A /, P B /6, and P C /. What is P A B C C? Solution: Thus P A B C P A B P AP B. A B B C B C A B C A B C C C. P A B C C P A B C + P C P A B C C [ P AP B] + P C P C points Let P A 0. and P B A 0.. For what value of P B are A and B independent? Remember B A def {ω B : ω / A}. Solution: Since B B A A B is a disjoint union: P B P A B+P A B 0.+P A B 0.+P AP B + P B, Which implies P B or P B 6.

3 Probability and Statistics I Sample Exam #. 0 points A box contains three cards. One card is red on both sides, one card is green on both sides, and one card is red on one side and green on the other side. One card is selected from the box at random and the color green on one side is observed. What is the probability the other side of this card is also green? Solution: Let {a, b} be the sides of card both sides red. {a, b} be the sides of card both sides green. {a, b} be the sides of card a red and b green. Define G A def {a, b, b} the card face chosen was green def {a, b} the second card was chosen Then A G {a, b} P A G P A G P G /6 / points There is a box with five balls in it. Three of the balls are blue and two are red. Each time you pull out a red ball a new one is added. This means that every time you reach into the box there are two red balls in the box. The number of the blue balls may change, but not the number of red ones. The box is closed and you cannot see which ball you will pull out. Assume that you are equally likely to pull out any of the balls in the box. You draw three balls from the box and let X be the number of blue balls you pulled out. Find the density of X.

4 Probability and Statistics I Sample Exam # Solution: Outcomes: Thus outcome X P B bbb 0 f X x bbr brb rbb brr rbr rrb rrr if x if x if x 0 if x points sample Roll a four sided, fair die twice. If i is the outcome for the first roll and j is the outcome of the second roll, what is the density of X maxi, j? Solution: Listing all the equal likely outcomes of rolling the die twice:, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X

5 Probability and Statistics I Sample Exam # Thus if x 6 if x 6 f X x if x. 6 7 if x 6 8. Data for the graduates of Macademia High School class of 007 is given below. Given a graduate, ω, one lets Xω # years of post secondary education graduate ω has and Y ω # hours per day rounded off to nearest hour of tv, graduate ω views Find the marginal distribution of Y. X Y Solution: Consider, Now note that X Total Y Total if y 0 if y 8 f Y y if y. if y 0

Probability and Statistics I Sample Exam # 9. 0 points Let X and Y be continuous random variables with joint density function { x+y for 0 < x < and 0 < y < x f XY x, y. What is P X >?

6 Probability and Statistics I Sample Exam # 9. 0 points Let X and Y be continuous random variables with joint density function { x+y for 0 < x < and 0 < y < x f XY x, y. What is P X >? Solution: {x,y:x>} f XY x, y da x + y da S x 0 x + y dy dx xy + y x dx ] y x y0 ] x x [ [ ] ]. dx 0. 0 points In the above problem, find the marginal distribution, f Y y of f XY x, y.

7 Probability and Statistics I Sample Exam # Solution: f Y y { y x+y dx if 0 y 0 { y +y+ if 0 y 8.

The probability of an event is viewed as a numerical measure of the chance that the event will occur.

Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that