Answer the following questions: 1. Find the probability mass function of a random variable X corresponding to the absolute value of the difference between the results of two dice. 2. Find the value of...

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Answer the following questions: 1. Find the probability mass function of a random variable X corresponding to the absolute value of the difference between the results of two dice. 2. Find the value of c if the probability mass function of a random variable X is: 3. Suppose 10 fair coins are tossed. Find the probability there will be more heads than tails. 4. Consider a sphere of radius r centered at (0, 0, 0). Express the following sets in set notation. (a) The points on the surface of the sphere. (b) The points inside the sphere. (c) The points outside the sphere. (d) The points inside the sphere but outside the sphere of radius r/2 centered at (0, 0, 0). (e) The points inside the sphere but outside the cube of length 3r/2 centered at (0, 0, 0). 5. Suppose we have these points: A = (a1,a2) , B = (b1,b2) C = (c1,c2) , D = (d1,d2) where a1 < b1="">< c1="">< d1="" and="" a2="">< b2="">< c2="">< d2. **note: the path only moves right or up on the grid. (a) give an expression for the probability that a random path from a to d will pass through both b and c. (b) give an expression for the probability that a random path from a to d will pass through b, but not c. (c) give an expression for the probability that a random path from a to d will pass through neither b nor c. 6. find the probability mass function of a random variable x corresponding to the absolute value of the difference between the results of two dice. 7. suppose 10 fair coins are tossed. find the probability there will be more heads than tails. 8. suppose that a woman is equally likely to have a male or female child. a couple wishes to have exactly three female children in their family. they will have children until this condition is fulfilled. (a) what is the probability that the family has k male children? (b) what is the probability that the family has four children? (c) what is the probability that the family has at most four children? 9. suppose archana and katie play a game where they alternate (in this order) in rolling two fair dice until the first person rolls a sum of 8 on a single turn, who wins the game. (a) find the probability that each person wins the game. (b) how much of an advantage does archana have? (c) suppose archana goes first, then katie gets k turns, and then they alternate. how large must k be so that katie is more likely to win? 10. three dice are rolled. if each land on a different number, find the probability that one is 3. 11. (optional) write an r script to simulate tossing a coin. plot the evolution of the relative frequency of heads in a sequence of 100 coin tosses 12. a geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. the geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis. (a) what is the pmf of the number of granite specimens selected for analysis? (b) what is the probability that all specimens of one of the two types of rock are selected for analysis? d2.="" **note:="" the="" path="" only="" moves="" right="" or="" up="" on="" the="" grid.="" (a)="" give="" an="" expression="" for="" the="" probability="" that="" a="" random="" path="" from="" a="" to="" d="" will="" pass="" through="" both="" b="" and="" c.="" (b)="" give="" an="" expression="" for="" the="" probability="" that="" a="" random="" path="" from="" a="" to="" d="" will="" pass="" through="" b,="" but="" not="" c.="" (c)="" give="" an="" expression="" for="" the="" probability="" that="" a="" random="" path="" from="" a="" to="" d="" will="" pass="" through="" neither="" b="" nor="" c.="" 6.="" find="" the="" probability="" mass="" function="" of="" a="" random="" variable="" x="" corresponding="" to="" the="" absolute="" value="" of="" the="" difference="" between="" the="" results="" of="" two="" dice.="" 7.="" suppose="" 10="" fair="" coins="" are="" tossed.="" find="" the="" probability="" there="" will="" be="" more="" heads="" than="" tails.="" 8.="" suppose="" that="" a="" woman="" is="" equally="" likely="" to="" have="" a="" male="" or="" female="" child.="" a="" couple="" wishes="" to="" have="" exactly="" three="" female="" children="" in="" their="" family.="" they="" will="" have="" children="" until="" this="" condition="" is="" fulfilled.="" (a)="" what="" is="" the="" probability="" that="" the="" family="" has="" k="" male="" children?="" (b)="" what="" is="" the="" probability="" that="" the="" family="" has="" four="" children?="" (c)="" what="" is="" the="" probability="" that="" the="" family="" has="" at="" most="" four="" children?="" 9.="" suppose="" archana="" and="" katie="" play="" a="" game="" where="" they="" alternate="" (in="" this="" order)="" in="" rolling="" two="" fair="" dice="" until="" the="" first="" person="" rolls="" a="" sum="" of="" 8="" on="" a="" single="" turn,="" who="" wins="" the="" game.="" (a)="" find="" the="" probability="" that="" each="" person="" wins="" the="" game.="" (b)="" how="" much="" of="" an="" advantage="" does="" archana="" have?="" (c)="" suppose="" archana="" goes="" first,="" then="" katie="" gets="" k="" turns,="" and="" then="" they="" alternate.="" how="" large="" must="" k="" be="" so="" that="" katie="" is="" more="" likely="" to="" win?="" 10.="" three="" dice="" are="" rolled.="" if="" each="" land="" on="" a="" different="" number,="" find="" the="" probability="" that="" one="" is="" 3.="" 11.="" (optional)="" write="" an="" r="" script="" to="" simulate="" tossing="" a="" coin.="" plot="" the="" evolution="" of="" the="" relative="" frequency="" of="" heads="" in="" a="" sequence="" of="" 100="" coin="" tosses="" 12.="" a="" geologist="" has="" collected="" 10="" specimens="" of="" basaltic="" rock="" and="" 10="" specimens="" of="" granite.="" the="" geologist="" instructs="" a="" laboratory="" assistant="" to="" randomly="" select="" 15="" of="" the="" specimens="" for="" analysis.="" (a)="" what="" is="" the="" pmf="" of="" the="" number="" of="" granite="" specimens="" selected="" for="" analysis?="" (b)="" what="" is="" the="" probability="" that="" all="" specimens="" of="" one="" of="" the="" two="" types="" of="" rock="" are="" selected="" for="">