Please work out and explain! Thank you! Let a discrete random variable N be geom

Question

Please work out and explain! Thank you!

Let a discrete random variable N be geometrically distributed: N- p(1-p)" for n -1,2,3 3,... Let N be the number of trials that are made until a success occurs (with probability p. Suppose you are rolling a true die and a success is having a 1 or a 2 occur (dots on the top face of the die) (i) What is the probability that this will take exactly 1 roll? (ii) Given that you have made 2 unsuccessful rolls, what is the probability that you will succeed on the next roll? (iii) Given that you have made 4 unsuccessful rolls, what is the probability that you will succeed on the next roll?

Explanation / Answer

Here, the probability of success (getting 1 or 2) is

p = 2/6 = 1/3

Hence,

i)

P(1) = p (1-p)^(1-1) = p = 1/3 [ANSWER]

**************

ii)

The rolls are indepdendent. Hence, the success on the next roll is still

p = 1/3 [ANSWER]

********************

iii)

The rolls are indepdendent. Hence, the success on the next roll is still

p = 1/3 [ANSWER]

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