From noon until sunset, bees emerge from a beehive according to a Poisson proces

Question

From noon until sunset, bees emerge from a beehive according to a Poisson process with rate =2 per second.

a.Find the probability that bees 1 and 2 emerge less than 0.25 seconds apart while bees 3 and 4 emerge more than 0.25 seconds apart. .024

b.Find the probability that bees 5 and 6 emerge less than 0.25 seconds apart while bees 7 and 10 emerge more than 0.25 seconds apart. .0387

c.Find the probability that the 2nd bee emerges less than 1.05 seconds after noon and the 4rd bee emerges more than 1.05 seconds after noon. (Hint: Are the events independent?)

Explanation / Answer

here =2/sec

hence a)probability that bees 1 and 2 emerge less than 0.25 seconds apart while bees 3 and 4 emerge more than 0.25 seconds apart =(1-e-t1)*(e-t1) where t1=0.25

=(1-e-2*0.25)*e-2*0.25 =(0.3935)*(0.6065)=0.239

b)probability that bees 5 and 6 emerge less than 0.25 secondsand 7 and 10 emerge more than 0.25 seconds apart =(1-e-t1)*(e-t2* (t2)x1/x1!+e-t2* (t2)x2/x2!+e-t2* (t2)x3/x3!) where t1=0.25; t2=0.25';x1=0 ;x2=1;x3=2

=(0.3935)*(0.6065+0.3033+0.0758)=0.3578

c) probability that the 2nd bee emerges less than 1.05 seconds after noon and the 4rd bee emerges more than 1.05 seconds after noon =P(2=<X<=3) =0.4553

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