Probability of taking cycle rickshaw It is known that IIT bus will arrive at random at Nilgiri hostel bus stop sometime between $8:30$ AM and $8:45$ AM. Rahul decides that he will go at random to this location between these two times and will wait at most $5$ minutes for the bus. If he misses it, he will take the cycle rickshaw. What is the probability that he will take the cycle rickshaw? Answer- $13/18$. This question was asked in my exam. I solved it pretty weirdly and got the answer $4/9$. My solution:- Probability of him taking the bus:- $1/9$(when he reaches between 8:30 and 8:35 and the bus also comes during that interval)+$1/9$(Similar, but with interval 8:35-8:40)+$1/3$(When he reaches between 8:40-8:45. I need not consider the bus since if it hasn't arrived till now, he'll definitely catch it)$=5/9$ Where am I wrong? OR Can you please provide a better solution? Thanks in advance!

The bus and Rahul do not have to arrives at the same $5$ minutes interval that you described. Also, even if it is the same interval, Rahul might reach later than the bus.

Let $R$ be arrival time of Rahul and $B$ be arrival time of $B$. To take the bus, Rahul need to be earlier than the bus by at most $5$ minutes.

We are interested in

\begin{align}1- Pr( B -5 \le R \le B)&=1-\left(\frac12 - \frac12\left( \frac23\right)^2\right)\\\ &=\frac12 +\frac12\cdot \frac49\\\ &=\frac12\frac{13}{9} \\\ &=\frac{13}{18} \end{align}

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