Mosquito Larvae!

Probability Level 2

A 5 ltr bucket of water is taken from a swamp. The water contains 75 mosquito larvae. A 200mL flask of water is taken from the bucket for further analysis. What is the probability that the flask contains at least one mosquito larvae?

0.950 0.999 1 0.825

1 solution

Maruf Islam Joy
Dec 14, 2019

We will solve this problem with Poisson Distribution. So we need to find λ. We know λ=n(no of trials, here no of larvae) P =75 $\frac{200}{5*1000}$ =3. We have to find the probability fo at least one that means the probability for 1 or above case. So, we can calculate this using Poisson Distribution. P(x>=1)=1-P(x<1)=1-P(x=0)=1-((e^-λ ) λ^x)/x!=1-((e^-3 ) 3^0)/0!=0.950

Why the Poisson distribution? Assuming that the probability that a larva is in any particular region of water is proportional to its volume, the probability that a mosquito is in the smaller flask is $\tfrac{1}{25}$ . Thus the probability that none are in the smaller flask is $\left(\tfrac{24}{25}\right)^{75}$ , making the probability that at least one is in the smaller flask equal to $1 - \left(\tfrac{24}{25}\right)^{75} = \boxed{0.953}$ You are using the Poisson approxiation $\mathrm{Po}(3) \sim B(75,\tfrac{1}{25})$ . Since the calculation with the Binomial distribution is easy to do, the Binomial calculation is more appropriate.

Mark Hennings - 1 year, 5 months ago

I agree, the Binomial distribution gives the actual answer.

Jon Haussmann - 1 year, 5 months ago

Thanks, Mark Sir. Yeah, it is

Maruf Islam Joy - 1 year, 5 months ago

Mosquito Larvae!

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