[bushindo]

In a certain bacterial colony, all bacteria reproduce by asexual reproduction. That is, every bacterium over the span of its life has a 1/2 chance of splitting into 3 identical clones. The clones then have the same reproductive mechanism as their parent. If a bacterium doesn't divide (1/2 chance), it will die eventually.

Assuming that the bacterial company starts with just two bacteria. Assuming that the division process for the bacteria are independent of each other, what is the chance that the colony would go extinct, i.e. having all the bacteria dead at some point?

[Guest]

In a certain bacterial colony, all bacteria reproduce by asexual reproduction. That is, every bacterium over the span of its life has a 1/2 chance of splitting into 3 identical clones. The clones then have the same reproductive mechanism as their parent. If a bacterium doesn't divide (1/2 chance), it will die eventually.

Assuming that the bacterial company starts with just two bacteria. Assuming that the division process for the bacteria are independent of each other, what is the chance that the colony would go extinct, i.e. having all the bacteria dead at some point?

My answer:

This is a probability series multiplication problem. I came up with the formula:

multiply from i=2 to n (1-1/2^n)

For the first split, the odds of splitting is 1-1/4 (or 75%)

Assuming at least 1 split, the odds of splitting again now that there are 3) are 1-1/8

and so on...

so:

(3/4)*(7/8)*(15/16)*(31/32)....

comes out to about 58% chance of survival

[Guest]

i would think it would just be

i would think it would just be .5^(4k+2)*(4k+2)Choosek/2/((k!)*(3k+2!))

summation over k=0to bazillion

This probably simplifies to .5 but i don't know i really haven't looked.

but that looks annoying and i haven't really looked at the problem that much so ignore me for now. Im just writing this so in the case that im right you can all watch in amazement. but in the case im wrong i clearly wrote i haven't looked at the problem so its really a win win. Unlike the penguins game tonight.

[Guest]

Suppose that we have a single bacteria A in the colony. Let p(A) be the probability that, after some amount of time, all descendants of A are dead. Call this event E, for extinction. In terms of p(A), the answer to the question would be p(A)² since the two starting bacteria are identical and divide independently of one another.

To find p(A), there are essentially two ways that E can occur. Either (1) the original bacteria fails to divide (with probability 1/2) or (2) A divides into three bacteria a,b and c, and all of their descendants eventually die out. Since all the bacterias are identical, the probability that bacteria a's descendants die out is p(a). Possibility (2) occurs with probability (1/2)*p(a)*p(b)*p( c).

Possibilities (1) and (2) are disjoint events so p(A)=(1/2)+(1/2)*p(a)*p(b)*p( c). Again, since all bacteria are identical, the probabilities are identical: p(A)=p(a)=p(b)=p( c)=p. So p=(1/2)+(1/2)*p³. The positive real solutions to this equation are (5^(1/2)-1)/2 and 1.

The question then becomes: does the colony survive indefinitely with a non-zero probability? If yes, then p(A)=(5^(1/2)-1)/2. I would say yes, but at this stage I don't know why it would be so. Can someone else say why?

I thought that this question was a good one, and very interesting.

[Guest]

In a certain bacterial colony, all bacteria reproduce by asexual reproduction. That is, every bacterium over the span of its life has a 1/2 chance of splitting into 3 identical clones. The clones then have the same reproductive mechanism as their parent. If a bacterium doesn't divide (1/2 chance), it will die eventually.

Assuming that the bacterial company starts with just two bacteria. Assuming that the division process for the bacteria are independent of each other, what is the chance that the colony would go extinct, i.e. having all the bacteria dead at some point?

My answer: 25% chance. What's the answer please, if you would be so generous with your knowledge, honorable Bushindo.

powermonkey

[Guest]

hey manwe i agree with your findings except for a lil

i may be wrong but i think you have to square you answer you found the probability that one dies out

to my understanding the colony will survive forever with about .62 probability because the formula doesn't include outside influence. Just like the human species will live forever except for the sun burning out and the world ending in 2012 (the mayans said so)

any species that has an average off spring of 2 or more per pair (or 1 or more for asexuals) will live forever we just had to find the probability that that didn't happen

in my mind you have half a chance to gain two half a chance to lose one seems to me its an easy guess that they'll survive around 2/3 of the time but 20-20 hind sight i guess

[bushindo]

hey manwe i agree with your findings except for a lil

i may be wrong but i think you have to square you answer you found the probability that one dies out

to my understanding the colony will survive forever with about .62 probability because the formula doesn't include outside influence. Just like the human species will live forever except for the sun burning out and the world ending in 2012 (the mayans said so)

any species that has an average off spring of 2 or more per pair (or 1 or more for asexuals) will live forever we just had to find the probability that that didn't happen

in my mind you have half a chance to gain two half a chance to lose one seems to me its an easy guess that they'll survive around 2/3 of the time but 20-20 hind sight i guess

Good job to manwe and final. You got it. I agree with final that the chance of a colony living indefinitely is the complement of the chance of extinction. So, in this case, the two bacteria and their possible descendant will go extinct with probability 0.618034^2 = 0.381966. That means they'll exist indefinitely with probability 1- 0.381966 = 0.618034.

It is also interesting to note that every generation with k descendants will have an extinction chance of 0.618034^k. Of course, after a certain k, that chance is trivial.

Edited June 7, 2009 by bushindo

[Guest]

yah give it to manwe, i was doing something like what he was doing but ended up looking at his spoiler before i finished. I prolly would have gotten there eventually but it doesnt change the fact that I got help from his thought process . but hey cheating worked in highschool so y not here.

good puzzle I didn't expect the trick after the labor intensive problems of late

[Guest]

to my understanding the colony will survive forever with about .62 probability because the formula doesn't include outside influence. Just like the human species will live forever except for the sun burning out and the world ending in 2012 (the mayans said so)

or some klutz dropping the petri dish! Perhaps prophesied as the coming of the great white lab coat?