A bus will arrive at a uniformly distributed time over the interval 0 to 1/2 hour. A passenger arrives at the bus stop at a uniformly distributed time over the interval 0 to 1/4 hour. Assume that the arrival times of the bus and passenger are independent of one another. Let B be the time the bus arrives and P be the time the passenger arrives.

I am completely stumped on this problem. My professor said to draw out a graph and use an area argument since the arrivals are independent and uniform. However, even after drawing out the graph I'm still not understanding how everything is working. He made the X axis the bus, and the Y axis the passenger, forming a rectangle. He said that the line X = Y cuts through the rectangle, and the probability that the passenger arrives first is represented below the line X = Y. He then basically just multiplied the area under the line by 8.

I'm really not understanding the whole line X = Y portion, or why the area below the line represents the probability the passenger arrives before the bus. Is there another way to do this that doesn't make use of a graph?


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