Red car or blue car next?

Question

Red cars pass a highway viewpoint as a Poisson process at rate 20 per hour, and blue cars pass independently at rate 30 per hour. What is the probability that the next car to pass is blue?

Accepted Answer

The red and blue car streams are independent Poisson processes, and we want the blue stream to produce the next event. Its win-probability is its share of the total rate: P(	ext{blue first}) = \frac{\lambda{	ext{blue}}}{\lambda{	ext{red}} + \lambda{	ext{blue}}} = \frac{30}{20 + 30} = \frac{30}{50} = 0.6. A 60\% chance. All that matters is the 20\!:\!30 ratio of rates. Takeaway. Only the ratio of rates decides the race; blue at 30 versus red at 20 makes the next car blue with probability 	frac{30}{50} = 0.6.

Red car or blue car next?

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