Stats 125 Open Book Test April 2022
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Stats 125 Open Book Test
April 2022
1. Nina has a fair eight-sided die with sides labelled “1”, “2” and “3” .
Let x be the number that is uppermost when Nina rolls the die. Then, x has the probability mass function:
,, 0.375 if α = 1,
,
,
,
,
0.5 if α = 2,
jx (α) = ．
0.125 if α = 3,
,
,
,
,
,( 0 otherwise.
(a) How many sides on the die are labelled“1”? How many are labelled “2”? How many are
labelled “3”?
(b) Write down the cumulative distribution function of x , Fx (α).
(c) Find P(x < 3).
(d) Find Var(x).
(e) Find E(4 · 2x) and Var(4 · 2x).
(f) Let y be the number of times Nina rolls a “1” or “3” before she rolls the eighth “2” .
State the most appropriate distribution to use to model y and the parameter(s) of the distribution.
(g) Find E[y].
(h) Let Z be the number of times Nina rolls a “3” in 18 rolls of the die. State the most
appropriate distribution to use to model y and the parameter(s) of the distribution.
(i) Nina rolled the die six times and a “1” was rolled exactly once. Let w be the roll on which a “1” was rolled. State the most appropriate distribution to use to model w and the parameter(s) of the distribution.
(j) Find Var[w].
(k) Let v be the number of times Nina rolls a “1” or “2” before she rolls a “3” . State the
most appropriate distribution to use to model v and the parameter(s) of the distribution.
2. For the purposes of this question we define a 2010student to be a student with a student loan who left study in 2010. Consider the following information about 2010students:
20,637 2010students had completed their qualification and 30,639 had not completed their qualification. Of those that had completed their qualification 33% repaid their student loan within five years. Of those that had not completed their qualification 21% repaid their student loan within five years.
. Let ○ be the event that a randomly chosen 2010student had completed their qualification.
. Let L be the event that a randomly chosen 2010student repaid their loan within five years.
(a) Based on the above information, write down P(L l ○), P(L l ○c ), P(Lc l ○), and P(Lc l ○c ).
(b) What is the probability that a randomly chosen 2010student had completed their
qualification?
(c) What is the probability that a randomly chosen 2010student had repaid their student loan within five years?
(d) Suppose that half of the 2010students had completed their qualification, and half had not. What effect would this change have on the probability that a randomly chosen 2010student had repaid their student loan within five years? You may assume that the percentage in each group who had repaid their student loan is unchanged from the information above.
3. A chocolate manufacturer produces Easter chocolate boxes where each box contains a large Easter egg and five small chocolate bars, some dark chocolate and some milk chocolate. 10% of boxes have only one dark chocolate bar, 30% have three dark chocolate bars, and 60% have four dark chocolate bars.
Note: For this question you may also assume that the number of dark chocolate bars in one box is independent of all the other boxes. Also suppose that when someone takes out a chocolate bar they are equally likely to pick any of the chocolate bars in the box.
● Ali has been given one of these Easter chocolate boxes.
Let Bi be the event that the box Ali chooses has i dark chocolate bars in it. Let D be the event that Ali takes a dark chocolate bar out of the box.
(a) What is P(D)?
(b) Suppose that the first chocolate bar that Ali pulls out of the box is a dark chocolate bar.
What is the probability that this was the only dark chocolate bar in the box?
● Peter buys five of these Easter chocolate boxes.
(c) What is the probability that neither of Peter’s boxes contains four dark chocolates?
(d) What is the probability that at least one of Peter’ boxes contains four dark chocolates?
● Lisa buys five of these Easter chocolate boxes. Assume that exactly three of Lisa’s chocolate boxes are containing four dark chocolate bars. Lisa gives two of her five boxes to her brother Thomas. Let x be the number of Thomas’s boxes that have four dark chocolates in them.
(e) State the distribution of x including the parameter(s).
(f) What is the probability that neither of Thomas’ boxes contain four dark chocolate bars?
4. Let A, B, C be events such that P(A), P (B l A) and P(A n B) are all non-zero. Show that
P (B n C l A)
P (B l A)
2022-08-15