Two reactions both complete in a second

Question

Two chemical reactions begin at the same instant and proceed independently. Reaction A completes at an exponential rate of 1 per second. Reaction B completes at an exponential rate of 3 per second.

What is the probability that both reactions have completed within 1 second?

Accepted Answer

"Both completed within 1 second" is about the slower reaction, the maximum completion time. The reactions are independent, so the joint chance is the product of the two individual complete-by-1-second chances. For an exponential with rate \lambda, the chance of completing by time t is 1 - e^{-\lambda t}. With t = 1 second: Reaction A (rate 1): \;1 - e^{-1 \cdot 1} = 1 - e^{-1} \approx 0.6321. Reaction B (rate 3): \;1 - e^{-3 \cdot 1} = 1 - e^{-3} \approx 1 - 0.0498 = 0.9502. Multiply: P(	ext{both done by 1 s}) = (1 - e^{-1})(1 - e^{-3}) \approx 0.6321 	imes 0.9502 = 0.6006 \approx 0.601. Th…

Two reactions both complete in a second

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