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