Question

A kind of algae is distributed in water according to a PPP (Poisson Point Process). It is known that the expected number of bacteria is 2 per liter. Samples of water are provided in bottles with volume 20 cc.

If two bottles have altogether 10 algae, what is the probability that the one of them contains less
than 3 algae?

Answer 1

I've just started learning about all the different types of distributions. I'm not really sure which one would apply in this case

Answer 2

me neither but it looks like a conditional probability, so maybe it would help to know the probability that 40 cc contain 10 algae

Answer 3

which seems to me extremely unlikely, since the average is 2 per liter, but i guess we can compute it

Answer 4

if I already know that it does, would the probability of that occurring help?

Answer 5

can I just say there is a 0.5 chance the algae could have gone in either bottle?

Answer 6

well i am really not sure, but this is a conditional probability, so we have to divide by that number i think
i should be quiet, because i am not sure how to approach this question.
we know it is poisson with \(\lambda =2\) but that is per liter

Answer 7

since I already know that there are 10 altogether amongst the 2

Answer 8

This is actually part d of the problem. In earlier parts I did use the poisson formula and also the binomial distribution

Answer 9

yikes

Answer 10

i am not sure why this is not binomial as a matter of fact, because really you just know there are 10 in two bottles, and presumably they are randomly distributed among the two

Answer 11

i gotta run, back soon

Answer 12

ok

Answer 13

A kind of algae is distributed in water according to a PPP (Poisson Point Process). It is
known that the expected number of bacteria is 2 per liter. Samples of water are provided in bottles with volume 20 cc.

a) Find the probability that a bottle contains more that 3 algae.
b) Find the probability that 3 selected bottles have altogether more than 3 algae.
c) Find the probability that if bottles are tested in a row, we observe a bottle with less than 4 algae for the third time on the 20th test.
d) If two bottles have altogether 10 algae, what is the probability that the one of them contains less than 3 algae?
e) Find the probability that a bottle with more than 2 algae in it has exactly 5 algae.

Answer 14

that's all the parts to the problem if it helps any. I think I have the first 3 parts. I just need help on d

Answer 15

The Poisson Distribution is wonderfully scalable. 2 / litre ==> 0.04 / 20 cc

\[p(0\;in\;20\;cc) = \frac{0.04^{0}\cdot e^{-0.04}}{0!}\;=\;0.9607894392\]
\[p(1\;in\;20\;cc) = \frac{0.04^{1}\cdot e^{-0.04}}{1!}\;=\;0.0384315776\]
\[p(2\;in\;20\;cc) = \frac{0.04^{2}\cdot e^{-0.04}}{2!}\;=\;0.0007686316\]

Answer 16

Are you SURE it's only 2 / litre? Seems a little low.

Answer 17

that's what it says
I did P(X>3) = 1-P(X<=3) for the first part

Answer 18

If there are 10, total, the ONLY possible combinations are:

0,10
1,9
2,8
3,7
4,6
5,5
6,4
7,3
8,2
9,1
10,0

What is the combined probability of the first three and the last three?

Answer 19

6/11?

Answer 20

I was thinking there would be more to it than that. Something dealing with conditional probability and the poisson distribution

Answer 21

I'm not really sure though

Answer 22

No. They are not equally probable. You'll have to calculate the individual probabilities.

Answer 23

I don't understand, if they are equally probably then what am I calculating?