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RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES

AND STATISTICS

INTRODUCTORY MATHEMATICAL STATISTICS PRINCIPLES OF MATHEMATICAL STATISTICS

(STAT2001/6039)

Assignment 1 (2018 Semester 1)

Your solutions to this assignment should be placed in the appropriate box in the RSFAS School foyer by the due time and date (as provided in the Course Outline and on Wattle). Attach a cover sheet (as provided on Wattle) which has your ANU ID number. The assignment is out of 100 and is worth 10% of your overall course mark.

Problem 1 (Total 20 marks)

Suppose that P AB( ) = 0.1, P(B ? =C) 1 and P B( - =A) P A( ? =B) 2 (P A- B). Find P AB P AC P ABC P A P( )+ ()- ()- ( )+ ( )B -PC P ABC( )- ( ), or determine the range of possible values for this quantity if it cannot be calculated as a single number.

Problem 2 (Total 30 marks)

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446, if k = 1). The last person to roll wins the game. (E.g., if k = 2, then each of the sequences 446, 64445 and 465443445 results in Homer winning.) Find the probability that Homer wins, for each k = 0, 1, 2, 3 and 99.

Problem 3 (Total 50 marks)

A stack of ten cards has four hearts (which are red), three diamonds (which are red), two spades (which are black) and one club (which is black). Four cards are sampled from the stack, randomly and without replacement, and placed in a box. Then one card is randomly selected from the box. If that card is a heart then all of the other three cards in the box are burnt. Otherwise, two cards are sampled, randomly and without replacement, from the other three cards in the box, and then these two cards are burnt. Find the probability that exactly k red cards are burnt, for each possible value of k.

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AND STATISTICS

INTRODUCTORY MATHEMATICAL STATISTICS PRINCIPLES OF MATHEMATICAL STATISTICS

(STAT2001/6039)

Assignment 1 (2018 Semester 1)

Your solutions to this assignment should be placed in the appropriate box in the RSFAS School foyer by the due time and date (as provided in the Course Outline and on Wattle). Attach a cover sheet (as provided on Wattle) which has your ANU ID number. The assignment is out of 100 and is worth 10% of your overall course mark.

Problem 1 (Total 20 marks)

Suppose that P AB( ) = 0.1, P(B ? =C) 1 and P B( - =A) P A( ? =B) 2 (P A- B). Find P AB P AC P ABC P A P( )+ ()- ()- ( )+ ( )B -PC P ABC( )- ( ), or determine the range of possible values for this quantity if it cannot be calculated as a single number.

Problem 2 (Total 30 marks)

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446, if k = 1). The last person to roll wins the game. (E.g., if k = 2, then each of the sequences 446, 64445 and 465443445 results in Homer winning.) Find the probability that Homer wins, for each k = 0, 1, 2, 3 and 99.

Problem 3 (Total 50 marks)

A stack of ten cards has four hearts (which are red), three diamonds (which are red), two spades (which are black) and one club (which is black). Four cards are sampled from the stack, randomly and without replacement, and placed in a box. Then one card is randomly selected from the box. If that card is a heart then all of the other three cards in the box are burnt. Otherwise, two cards are sampled, randomly and without replacement, from the other three cards in the box, and then these two cards are burnt. Find the probability that exactly k red cards are burnt, for each possible value of k.

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