# The New and Improved Lane

**Discrete Mathematics**Level pending

Lane had a problem with being late for his ﬁrst class of the day over his ﬁrst three years of high school. However, this year he has made a conscious eﬀort to improve.

On the ﬁrst day of the week, usually Monday, Lane’s probability of being on time for his ﬁrst class of the day is \(\dfrac { 2 }{ 3 } \). If he is on time on any particular day, the probability that he will be on time the next day is also \(\dfrac { 2 }{ 3 } \). However, if he is late on any particular day, the probability of being late on the next day is one-half the probability of being late on the previous day.

Lane attends school each of the ﬁve days from Monday to Friday one week. If the probability that he will be late for his ﬁrst class at least three times in that week can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

##### This problem has been proposed by the University of Waterloo.

**Note**: I committed a mistake solving the problem, so I'm posting it again. Obviously, the answer is not the same.