A combined experiment is performed by rolling a die with sides numbered from 1 t

Question

A combined experiment is performed by rolling a die with sides numbered from 1 to 6 and a child's block with sides labeled A through F. a) Write all of the elements of the cartesian product space. b) Define K as the event of obtaining an even number on the die and a letter of B or C on the block and find the probability of the event K. A combined experiment is perfomed by flipping a coin three times. a) Write all of the elements in the product space by indicating them as HHH, HTH, etc. b) find the probability of obtaining exactly two heads. c) Find the probability of obtaining more than one hea. Two men each flip a coin three times. a) What is the probability that both men will get exactly two heads each? b) What is the probability that one man will get no heads and the other man will get three heads? In playing an opponent of equal ability, which is more probable: a) To win 4 games out of 7, or to win 5 games out of 9? b) To win at least 4 games out of 7, or to win at least 5 games out of 9? A file containing 10,000 characters is to be transferred from one computer to another. The probability of any one character being transferred in error is 0.001. a) Find the probability that the file can be transferred without any errors. b) Using the DeMoivre-Laplace theorem, find the probability that there will be exactly 10 errors in the transferred file. c) What must the probability of error in transferring one character be in order to make the probability of transferring the entire file without error as large as 0.99?

Explanation / Answer

1-9.1 (a) There are 6 possiblities when rolling a die = 1,2,3,4,5,6

There are 6 possibilities while choosing an alphabet from child's block = A,B,C,D,E,F

There will be 6 X 6 = 36 elements in the cartesian sample space

sample space = (1,A), (1,B), (1,C),(1,D),(1,E),(1,F), (2,A), (2,B), (2,C),(2,D),(2,E),(2,F), (3,A), (3,B), (3,C),(3,D),(3,E),(3,F) ,(4,A), (4,B), (4,C),(4,D),(4,E),(4,F), (5,A), (5,B), (5,C),(5,D),(5,E),(5,F) ,(6,A), (6,B), (6,C),(6,D),(6,E),(6,F)

(b) K = Even number on die or Letter of B or C on the block.

K = (2,B) , (2,C) , (4,B) , (4,C) , ( 6,B) , (6,C)

Probability of event K = 6/ 37 = 1/6

1- 9.3 Coin flip three times

(A) Sample space = (HHH) , (HHT), (HTT) , (HTH) , (TTT), TTH), (THH) , (THT)

(B) Pr ( Two Heads)

Sample space = (HHT) , (HTH) , (THH)

Pr(Two Heads) = 3/8

(C) Pr (More than 1 heads) = Pr(2 Head) + Pr( 3 HEad)

= 3/8 + 1/8 = 4/8 = 1/2

1- 10.1

(a) Lets say probability is p, equal ability so p = 0.5

so Pr( 4 win out of 7) = 7C4 (0.5)7 = 0.2734

Pr (5 win out of 9) = 9C5 (0.5)9 = 0.246

so winning 4 out of 7 games has more probability then winning 5 out of 9 games.

(b) win atleast 4 games out of 7

Pr( win atleast 4 games out of 7) = Pr(4 win) + Pr(5 win) + Pr( 6 win) + Pr( 7 win)

= BIN (x >= 4; 7; Cumulative = yes) = 0.5

Pr( win atleast 5 games out of 7) = Pr(5 win) + Pr(6 win) + Pr(7 win) + Pr( 8 win) + Pr(9 win)

= BIN (x >= 5; 9; Cumulative = yes) = 0.5

so it will be equal for both.

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