find probability of taking taxi between two places Tom knows that the bus will arrive at station at random between 7:00 am to 8:30 am. He will go at random to the station to get a bus to work and waits for 5 minutes, if he misses the bus, he will take taxi to work. what is the probability that tom takes taxi to work? (assume uniform distribution for bus arrival and tom arrival to the station)

Look at the visualization below. The $x$-axis represents the time Tom arrives at the station after 7:00 am. Similarly, the $y$-axis represents the time the bus arrives at the station after 7:00 am. The event ends at 8:30 am, 90 minutes after 7:00 am.

![Visualization](

The entire area describes all possible event because each point on the plane describes the time the bus arrives at the station and the time Tom arrives at the station respectively. Notice the black strip. That's when Tom is able to catch the bus because that is when the bus arrives within 5 minutes after Tom's arrival. Hence the probability of Tom taking the taxi is

$\frac{\text{white area}}{\text{area of the square}} = \frac{\frac{1}{2}\times 90\times 90 + \frac{1}{2}\times 85\times 85}{90\times 90} = \frac{613}{648}$.