This Bacteria Problem Breaks Intuition 🤯 #jeeadvanced

Added: 2026-05-10

[00:00:04]Hello everyone. This is another students quest.

[00:00:08]This question from probability is sent to me by Aditya Anand from Patna.

[00:00:15]The question says that a certain kind of bacteria either die, split into two, or split into three bacteria.

[00:00:24]Uh there should be bacterium. It means one single entity actually is has three uh possibilities to grow.

[00:00:33]The grow means the first is they no grow like either die.

[00:00:38]The second is split into two. It will grow into two.

[00:00:42]And split into three bacteria. So again it will split into three.

[00:00:46]And each uh bacteria is identical exact copies means like this is just uh suppose you are in a certain stage, you are thinking that this is a new stage or the stage as it is copied from the beginning.

[00:01:02]The chance of dying is 1 by 4. The chances of uh splitting into two is half and splitting into three is 1 by 4.

[00:01:10]If the probability that it survives for infinite length of time is m minus root 3 by n where m and n are natural numbers, so we have to find out the ratio of m and n.

[00:01:21]So before I uh solve this question, I request you to attempt this question.

[00:01:27]And then we will see what is happening.

[00:01:30]So I hope you attempted it. Let's try to do it. Suppose I have one bacterium at the initial stage and it may die. Die means uh actually there no growth in it. So suppose it dies, what is the chance that it will get survived for any lineage?

[00:01:49]It's zero.

[00:01:50]Now it dies, so the probability it is given as 1 by 4.

[00:01:54]Then it is split into two. So let us say that split into two.

[00:02:01]Then that chance is given 1 by two.

[00:02:06]And then split into three, then that probability is given one over four again.

[00:02:13]Now, let's try to understand it this way.

[00:02:15]At this stage, they no growth further.

[00:02:20]But here what will happen? Suppose this is two, it is split into two means from split into two means one thing happen, incident happen, split into two.

[00:02:31]Now, at this position, I have two such bacteria. Like one bacteria is here and one bacteria is here.

[00:02:40]And from this position you want to observe the entire scene or entire thing for this bacteria, again it's a new kind of same bacteria has happened. So, as if uh the growth is starting from here again.

[00:02:59]So, again this bacteria has two possibilities that it can Sorry, three possibilities. It can die here soon or it can split into two or it can split into three.

[00:03:11]Similarly for it. Again, if if it split into three, it means at this stage what is happening? That three such copy of the same bacterium.

[00:03:23]And at this position, again one entity or one copy has again three choices. Either it can die or it can split into two or it it can split into three.

[00:03:36]If it is splitting into two or three, there's a chance of survival. But at every stage, suppose it dies, it means there no such further growth is happening. And in that case, there no survival.

[00:03:51]So, it it's actually convenient to find out the dying probability or non-survival probability rather than survival probability.

[00:04:02]So, how we define this whole thing in language? Let's Let me just write this first. So, let us call that N is an event that uh for which the event of not survival uh at any stage.

[00:04:27]At any stage means at one particular stage I'm saying that from there uh or maybe at N stage or any at any particular stage.

[00:04:47]Means we are uh suppose sitting on one stage here and we try to look at that after that, what is the chance of getting non-survival?

[00:04:57]So, this is actually not at particular stage is not surviving, it's from any particular stage should be a correct word. So, let's call it from any particular stage.

[00:05:13]Now, the S is I'm calling the event that is no survival.

[00:05:20]Sorry, survival, sorry.

[00:05:23]So, survival from any particular stage and then by then I say that the probability of N plus probability of S must be one.

[00:05:33]And for convenient, let's say probability of N is small P.

[00:05:39]So, for just calculation purpose. So, let's think in this way now.

[00:05:45]Suppose I want to calculate the probability that uh it's a non-survival after death.

[00:05:58]So, this is actually probability of non-survival after death of the bacteria at any stage.

[00:06:14]So, at any stage it is dying or it just is dead, then what is the probability of a non-survival from that a stage if it dies?

[00:06:23]So, there is going to be probability of dying is 1 by 2.

[00:06:28]Let me confirm. Yeah, sorry, 1 by 4.

[00:06:31]Probability of dying is 1 by 4.

[00:06:34]And if it dies, what is the chance of it not surviving? Of course, it's one.

[00:06:41]So, this is going to be 1 by 4 into 1.

[00:06:43]Now, suppose you again want to define it, probability of a non-survival if it splits into two.

[00:06:54]T W O, let's say.

[00:06:55]So, I'm writing in this way, probability of non-surviving after a splitting into two.

[00:07:08]into two.

[00:07:10]So, when you write that, you notice that if I can say that here one probability of a splitting into two is 1 by 2.

[00:07:19]And suppose it is there, it has split into two there, and now for each copy of the bacterium, there's a new kind of beginning as if you are starting from the same beginning.

[00:07:35]It is like suppose you are on this a stage, you are thinking of non-survival for the bacteria as if you are in the beginning and again thinking the same thing.

[00:07:46]It means from here again for non-survival of the bacteria, I can directly write as the small p.

[00:07:55]Because this is the non-survival from for any stage from any stage actually.

[00:08:01]Now, suppose a But the problem in this situation is it is splitting it is a splitting into two.

[00:08:09]And splitting into two means that at this position after half, there's one bacteria here and one bacteria here. Suppose in this situation what is happening this gets survived and this get not survived. It means now this entire definition will be wrong. Why? Because we are saying that probability of no survival after splitting into two.

[00:08:37]It means both of the branches after this or both of the copies should not survive.

[00:08:41]And each copy is identical, so I can say that this is nothing but p into p. Each has no survival probability.

[00:08:49]Now, probability of three is split is split into three and no survival. So again you understand what the detail I'm going to write it.

[00:09:00]So probability of no survival after splitting into three.

[00:09:10]Into three.

[00:09:18]So that is going to be 1 by 4 and now you notice that there's a three pieces of same bacteria. So this has no survival, this has no survival, this has no survival.

[00:09:29]So that is going to be p into p into p.

[00:09:32]Now, if I want to say that no survival, then I say that the no survival total probability it must be equal to 1/4 plus this when it will split into two plus this when it is splits into three.

[00:09:53]So I can write there's a 1/2 p squared and then 1/4 p cubed.

[00:10:00]Now all that I have to find out the value of p from this cubic equation. So let's find it. So this is going to be 4p is equal to one plus sorry, yes, one plus 2p squared plus p cubed.

[00:10:18]Therefore p cubed minus 4p plus 2p squared plus 1 is equal to zero.

[00:10:27]Now we can apply the remainder what we call rational root theorem and then we see that one is a possible solution or not. So let's substitute p is equal to one. So we can identify that this is p is equal to one.

[00:10:40]If I substitute that is going to be 1 minus 4 minus 3 and this is 3. Therefore p is equal to one is one of the roots.

[00:10:48]So I can say that p minus one is a one of the factor. So this is going to be p squared and minus p squared uh I have 2p squared and yes. So minus p squared so plus 3p I need.

[00:11:03]And then minus 3p and minus one.

[00:11:06]So this is going to be Now there's one in- interesting thing is p is equal to one if you're looking for.

[00:11:13]What is the meaning of this? Is certain death.

[00:11:20]And when it happens when there's no growth, there's no split. It means the growth the growth is so slow that it's to be certain.

[00:11:30]But that is not actually they're asking.

[00:11:32]They're asking that there's a See the question carefully. It survives for infinite length of time.

[00:11:40]So, we are not looking for that. We are looking for such probability which will not survive for infinite length of time.

[00:11:47]And then we are subtracting from that value from one, then we will get the survive for infinite number of time.

[00:11:55]So, this is not going to be happen. So, let's find it this thing. P squared plus 3p minus 1 is equal to zero. So, P is equal to negative 3 plus or minus square root of 9 plus 4 divided by 2.

[00:12:13]So, that must give you P is equal to negative 3 minus root 13 by 2, which is a rejected actually because a negative quantity. And we have to also confirm that the probability we are getting must be lesser than the value we are getting must be lesser than one.

[00:12:30]So, minus 3 plus root 13. Root 13 is less than 4.

[00:12:36]And we're subtracting a number which is less than 4 and more than 3.

[00:12:41]Uh from that you're subtracting 3 and dividing by 2, so that must be less than one.

[00:12:47]But, the what is the probability of survival? So, let's say S or P, what I have written, PS.

[00:12:53]So, PS is nothing but 1 minus P.

[00:12:58]Let's calculate it. So, 1 minus minus 3 plus root 13 by 2.

[00:13:05]So, that is going to be 5 minus root 13 by 2.

[00:13:09]What they're asking? They're asking that So, notice that this is also M minus root 13 by N. And I have M and N value known.

[00:13:18]So, M is 5 and N is 2 and they're asking M plus N. No, M by N.

[00:13:22]So, they're asking M by N. So, that is going to be 5 by 2, which is 2.5. It is a numerical one then.

[00:13:32]So, this is the solution for this question.

[00:13:35]And this uh idea came from the recursion theory of probability. Means we can make a iteration make an iteration process in probability system also. And from there we create an equation, and from that equation we get the probability.

[00:13:54]We can also think in this way that suppose here I have taken the uh like a mean probability at the non-survival.

[00:14:03]We can also make survival, but a little bit cumbersome to think or maybe hard to think, but we can do it.

[00:14:09]Suppose the survival you are taking, this value will become zero, but each of this we have to write 1 - S the whole square like and 1 - S the whole cube.

[00:14:18]So, again in the terms of S it will come as the cubic, and from there we can get direct value of S.

[00:14:27]So, that's all for today. See you in the next video with some other interesting question. Till then, stay math active.