The unbiased dice is tossed until a number greater than 4 appear. What is the probability that an even number of tosses is needed?

Explanation

According to the question, number greater than 4 may appear in `2^(nd),4^(th),6^(th),...` trial. Probability of getting number greater than 4 in any trial `=2/6=1/3`
`therefore` Probability of not getting the number greater than 4 in any trial `=1-1/3=2/3` Let A, be the evetn the number greater than 4 occurs in the trial. Now, result of each trial is independent of earlier trials. So, required probability
`=P(A_(1)'nnA_(2))+P(A_(1)'nnA_(2)'nnA_(3)'nnA_(4)'nnA_(5)'nnA_(6))+...`
`P(A_(1)')P(A_(2))P(A_(1)')P(A_(2)')P(A_(3)')P(A_(4))+...`
`=2/3xx1/3+2/3xx2/3xx1/3+2/3xx2/3xx2/3xx2/3xx2/3xx1/3+...`
`(2/3xx1/3)/(1-2/3xx2/3)("Sun of infinite G.P.")`
`=2/5`