Question

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))

Answer 1

$prime\;numbers\;that\; we\;can\;obtain\;are\;2,3,5\\ \therefore p=0.5,\\ P(x)=0.5(0.5)^{x-1},x=0,1,...\\ Mode=1,\\ E(x)=1/p=1/0.5=2,\\ 𝑃(𝑋 > 𝐸(𝑋))=𝑃(𝑋 > 2)\\ =1-P(X\leq 2)=1-[P(X=1)+P(X=2)]\\ =1-[0.5(0.5)^{1-1}+0.5(0.5)^{2-1}]=0.25$