Answer to Question #154580 in Statistics and Probability for lee

Question #154580

Let 𝑋 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 𝑋

Find 𝐸(𝑋)

Evaluate 𝑃(𝑋 > 𝐸(𝑋))


1
Expert's answer
2021-01-11T18:10:50-0500

primeβ€…β€Šnumbersβ€…β€Šthatβ€…β€Šweβ€…β€Šcanβ€…β€Šobtainβ€…β€Šareβ€…β€Š2,3,5∴p=0.5,P(x)=0.5(0.5)xβˆ’1,x=0,1,...Mode=1,E(x)=1/p=1/0.5=2,𝑃(𝑋>𝐸(𝑋))=𝑃(𝑋>2)=1βˆ’P(X≀2)=1βˆ’[P(X=1)+P(X=2)]=1βˆ’[0.5(0.5)1βˆ’1+0.5(0.5)2βˆ’1]=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,\\ 𝑃(𝑋 > 𝐸(𝑋))=𝑃(𝑋 > 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}]=0.25


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