Question

Consider an experiment where a fair die is rolled repeatedly
until the first time a 3 is observed.

i) What is the sample space for this experiment? What is the
probability that the die turns up a 3 after i rolls?

ii) What is the expected number of times we roll the die?

iii) Let E be the event that the first time a 3 turns up is after
an even number of rolls. What set of outcomes belong to this event?
What is the probability that E occurs?

Answer #1

i)

sample space for this experiment is S ={1,2,3,4,5,..........} ; as x or number of trail on which 1st 3 appears can taken any values from 1 to infinity,

probability that the die turns up a 3 after i rolls =P(there is
no 3 in first i rolls) =(5/6)^{i}

ii)

expected number of times we roll the die =1/p=1/(1/6)=6

iii)

here set of outcomes belong to this event are S ={3,5,7,9,11,....}

hence
P(E)=(5/6)^{2}(1/6)+(5/6)^{4}(1/6)+(5/6)^{6}(1/6)+(5/6)^{8}(1/6)+........=(5/6)^{2}(1/6)/(1-(5/6)^{2})

=(25/216)/(11/36)=25/66

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b) Let S denote the number of rolls until you get a repeated
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A fair die is rolled repeatedly. Find the expected number of
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4. A pair of fair dice is rolled repeatedly. Calculate the
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