Let π be the number of times an ordinary fair die is rolled, up to and including the roll on which
the first prime number is obtained.
Find the mode of π
Find πΈ(π)
Evaluate π(π > πΈ(π))
primeβ βnumbersβ βthatβ βweβ βcanβ βobtainβ βareβ β2,3,5β΄p=0.5,P(x)=0.5(0.5)xβ1,x=0,1,...Mode=1,E(x)=1/p=1/0.5=2,π(π>πΈ(π))=π(π>2)=1βP(Xβ€2)=1β[P(X=1)+P(X=2)]=1β[0.5(0.5)1β1+0.5(0.5)2β1]=0.25prime\;numbers\;that\; we\;can\;obtain\;are\;2,3,5\\ \therefore p=0.5,\\ P(x)=0.5(0.5)^{x-1},x=0,1,...\\ Mode=1,\\ E(x)=1/p=1/0.5=2,\\ π(π > πΈ(π))=π(π > 2)\\ =1-P(X\leq 2)=1-[P(X=1)+P(X=2)]\\ =1-[0.5(0.5)^{1-1}+0.5(0.5)^{2-1}]=0.25primenumbersthatwecanobtainare2,3,5β΄p=0.5,P(x)=0.5(0.5)xβ1,x=0,1,...Mode=1,E(x)=1/p=1/0.5=2,P(X>E(X))=P(X>2)=1βP(Xβ€2)=1β[P(X=1)+P(X=2)]=1β[0.5(0.5)1β1+0.5(0.5)2β1]=0.25
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