InterviewSolution
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InterviewSolution
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A player tosses a coin and score one point for every head and two points for every tail that turns up. He plays on until his score reaches or passes n. `P_(n)` denotes the probability of getting a score of exactly n. Which of the following is not true ?A. `P_(100)gt2//3`B. `P_(100)lt2//3`C. `P_(100),P_(101)gt2//3`D. None of these |
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Answer» Correct Answer - C The scores of n can be reached in the following two mutually exclusive events: (i) by throwing a head when the score is `(n-1),` (ii) by throwing a tail when the score is `(n-2)` Hence `P_(n)=P_(n-1)xx1/2+P_(n-2)+1/2" "[becauseP("head")="(tail")=1//2"]"` `=1/2[P_(n-1)+P_(n-2)]" "(1)` ` impliesP_(0)+1/2P_(n-1)=P_(n-1)+1/2P_(n-2)` `" "("adding"(1//2)P_(n-1)"on both sides")` `=P_(n-2)+1/2P_(n-3)` `=P_(2)+1/2P_(1)" "(2)` Now, a score of 1 can be obtained by throwing a head at a single toss. Therefore, `P_(1)=1/2` And a score of 2 can be obtained by throwing either a tail at a single toss or a head at the first toss as well as second toss. Therefore, `P_(2)=1/2+((1)/(2)xx(1)/(2))=3/4` From Eq. (2), we have `P_(n)+1/2P_(n-1)=3/4+1/2((1)/(2))=1` `or P_(n)=1-1/2P_(n-1)` `or P_(n)-2/3=1-1/2P_(n-1)-2/3` `or P_(n)-2/3=-1/2(P_(n-1)(2)/(3))` `=(-(1)/(2))^(2)(P_(n-1)-(2)/(3))` `=(-(1)/(2))^(3)(P_(n-3)-(2)/(3))` `=(-(1)/(2))^(n-1)(P_(1)-(2)/(3))` `=(-(1)/(2))^(n-1)((1)/(2)-(2)/(3))` `=(-(1)/(2))^(n-1)(-(1)/(6))` `(-(1)/(2))^(n)1/3` `or P_(n)=2/3+((-1)^(n))/(2^(n))1/3=1/3{2+((-1)^(n))/(2^(n))}` Now, `P_(100)=2/3+(1)/(3xx2^(101))gt2/3` and `P_(101)=2/3-(1)/(3xx2^(101))lt2/3` `P_(101)lt2/3ltP_(100)` |
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