𝐁𝐀𝐂 𝟐𝟎𝟐𝟓 𝐂𝐚𝐦𝐞𝐫𝐨𝐮𝐧 : exercice de 𝗽𝗿𝗼𝗯𝗮𝗯𝗶𝗹𝗶𝘁𝗲́ corrige en direct

Ajouté : 2026-05-17

[00:00:17]Good evening everyone.

[00:00:19]So good evening Nellie and good evening Ariel.

[00:00:43]We'll start with a quick question while we wait for the others to connect. It would simply mean that we've already had to do a multiple-choice quiz on this. If you are given something like I raised to the power of 2026, what value will that give you?

[00:01:04]OK, we'll start at 8:10 PM as long as the others are logging on. Tonight, we're going to talk about probability mixed with geometry in space. A simple exercise that came up in the 2025 Baccalaureate C exam last year for the C series.

[00:01:35]Good evening friends.

[00:01:38]If someone tells you it has a power of 2026 while waiting for others to connect, can someone tell me what that means? How is it that I think the volume is good?

[00:02:05]OK, we'll start correcting our exercise at 8:10 PM.

[00:02:15]In the meantime, a quick reminder: what does i to the power of 2026 give you? What does i to the power of 26 - 1 give you? Well, Ariel - 1, why?

[00:02:45]We'll start correcting our exercise at 20.

[00:02:59]Ariel, why do you think that the power 2026 gives mo- How do you proceed?

[00:03:22]Correction in 4 minutes.

[00:03:31]OK.

[00:03:33]Apart from that, if you have any questions, feel free to ask them. We are weeks away from the baccalaureate exam and for 2 weeks it's a really important phase during which we concentrate. This is also a phrase that many will feel, causing stress, fear, and panic. So during these last phases, in order to avoid or actually deal with this, take the time to get organized. It is said for example every day you must know what you are doing if it is for example 2 hours of mathematics, 2 hours of finishing what you are revising always find the time to make a rather flexible program.

[00:04:11]For example, instead of simply telling yourself I will definitely revise from 8 to 10am or from 4 to 6pm, you tell yourself that the essential thing is that I do my 2 hours. If you can't do it from 4pm to 6pm, you could do it now, rather from 5pm to 7pm. The goal for this final phase would simply be to tell you what you will be following each day. And this is the phase during which we make final revisions. This is no longer the time to try to understand your entire program, to try to master everything that was required. We're no longer at that level.

[00:04:43]Now, let's sit down and ask ourselves, what can come up in math? What might come up in physics? What can come up in chemistry? What can come in ST? What might come next in computing, and so on. You look at past topics, you observe what came up and you ask yourself, if this question comes up, can I solve it? If this one comes instead, can I resolve it? You try it, you can take it. There are weeks, take the first week for example to do this. You look at the past steps, you look at the questions that come. Are you thinking, "Do I know how to solve this?" How do we go about solving this? And so on. Now the following week, you deal with a topic there, you work both on your knowledge and on time.

[00:05:26]What does that mean? If you are given a math problem, and a physics break, are you able to finish within the allotted time? If you can't love yourself, then can you get the maximum points because it's true the majority won't finish their test but others reach this level that even if they don't finish they can't easily get 15 or 16 because before starting their test they located the easiest exercise. I'm going to start there because I know that by starting there I can earn the maximum number of points and so on. So this is the final period, a really interesting final period, and really take the time to get organized. Make time to do sports too because you will feel like you are getting overwhelmed because there is a lot to take in, a lot to master. So we do a little bit of sport maybe twice a week, that's good. Running, transferring, that's important. Okay, let's start correcting our exercise. But quickly, when you are given something like this and asked to solve it, then Ariel told us the rest is to say what? This means that here you will simply remember that if I say i to the power of 4 it is i squared, the rate squared. So this will make -1 squared which will give me 1. As there is the power kone 1. Each time, no matter what power we are here, we will simply divide first the primes on board by 4. So here we have 5 - 20 which will make us go down with 2 which makes 0, we go down with the 6 uh it will go six times I think. That makes 24, there's 2 left over. But simply because there's 2 left over, sir, I can write it as i to the power of 4, all to the power of 506 * i to the power of 2. So this is 1 to the power of 506 x i squared, which is 1 to any power, which is 1.

[00:07:37]This is me, very good Ariel.

[00:07:40]Now we'll move on to our exercise. Probability is a chapter that confuses many people, but every time you are given probabilities before moving on to questions, always make sure you understand the concept. And also before moving on to this exercise for those who are taking this baccalaureate in July, I am talking in particular about countries such as Niger, such as Gabon, such as uh I think these are the states of the Comoros. There is a second section for preparing for the baccalaureate. After that, which ends at 30 months, we have a second section which will start from the beginning of June to July. So if you want to participate in this, just write to me on WhatsApp. So let's begin with our exercise. Whenever you have a chapter on probability, the first thing I always advise you to do is take the time to understand the statement. That being said, can I use arrangement or simply combination here? So that's what we're going to start doing here. So let 's take something very simple. We are told this. I want to understand my problem. We say I have one which contains four numbered tokens - 1, 0 and 1. So we have four tokens in our... pardon, in our bag. We all say it's discernible by touch. When I say all discernible by touch, it's actually to say that there is an equal probability because if I can't differentiate each ball with my hand when I want to make a choice, it means that in fact there is the same chance that I take the ball with the number -1 as I take the one with the number 0 or the one with the number 1. Now, we tell you this. We take a token from the sap and note its number X. So we'll represent it like something like this. We said that the first token I have, I'm going to call it X, that makes minus 001. We write X down and put it back in the bag. What does that mean? Since there are four balls to start with, that means if I shot one, it means there are four balls. So I actually have four possibilities for drawing my token. When I put it back in the bag and draw a second token, it has four balls. I shot first, I noted his humor. When I put it back, it means I still have four balls in my bag and I have to hold on a second time. So, the second one I'm going to draw, I'll still actually have four possibilities because I've put back what I took at the start. So now we note the second one we obtained, Y, and we put it back in the bag. That means I put that ball back in again and my bag always comes back to four balls. Good evening Séverine. My bag always comes back with four balls.

[00:10:23]Now we say, then we draw a third token. This means that since we still end up with four balls and we denote this as Z, it means that our Z will still have four possibilities. We say this: "For each draw of three tokens, we associate the space of a coordinate system Ojk with the point M at coordinates X, Y, Z. What does that mean? It simply means that if I take this point with coordinates X, Y, Z, it means that the token, or rather the number I get, corresponds to these coordinates. Since it's here, it means, for example, that if I drew -1, I might draw 0. I might even draw zero. So the first thing we'll notice here is that if I want the set of all possibilities, there are four choices for the first, four for the second, and four for the third. So here it would be 4 to the power of 3. So the P is full. We have four elements, and we repeat the same operation several times. There's the other element, and there's the repetition. So how many elements are there?" 4 and I want three elements. Now, let's move on. I'm told that a is the point with coordinates 1 - 1 - 1; it's a point in space. First question: we want to demonstrate that the probability that point M is A is equal to 1/64. Very good. We want to demonstrate that point M is A. We want this probability to be simply equal to 1/64.

[00:12:01]The first thing you're going to do here is simply go to your rough work. I was told that a has coordinates 1 - 1 - 1. So, if we're simply told that m coincides with a, what does that mean? I'd like to know what the x, y, and z values ​​mean.

[00:12:24]I'm listening. If m coincides with 1, what does that mean? What are the values ​​of X, Y, and Z?

[00:12:36]And let's make sure you're with me, guys. What are the values of X, Y, and Z? I'm listening.

[00:12:43]X Y Z.

[00:12:52][clears throat] H, what will the values ​​be for our first question of X and Z? If I'm told that A coincides with point M, do you have any questions for comprehension?

[00:13:14]I think so. Very good, man. So, very good, what does that mean? It means that X y Z is 1 - 1 and - 1. Very good.

[00:13:27]OK. X Y Z is 1 - 1 and -1. So far, folks, is that clear for P, the concept we're using? That is, here we've done it by order and repetition, we're using P. The cardinality of the universe is 4 to the power of 3. Did everyone understand that?

[00:13:54]Is that clear for 4 to the power of 3? Did everyone understand this?

[00:14:16]4 to the power of 3.

[00:14:26]Very good, let's go. So, what does that mean? X Y Z is 1 - 1 - 1. In other words, What was I told? I was told that my S will contain the balls with the numbers. We have -1, we have 0, and we have 0, 0, and 1. What do I want? I want my first ball to be 1, my second ball to be -1, and my last ball to be -1. Look, what would that say? It would simply say that the probability I'm looking for—I already know the probability is 4 to the power of 3. Now, my first ball is -1. That means that here, there's simply one ball that wants to take the value -1. It's true that when I make this first draw, there's only one possibility of getting the ball -1, sorry [clears throat] the ball 1. There's only one possibility. Now, I want the ball 1 and I want the ball -1. So when I take the ball with the number 1, there's only one I take a single ball and a second ball with the number -1. There's also only one ball. So it's also multiplied by 1. And I take a third ball with the number -1.

[00:15:39]Since when you take the second ball, you take it, note its negative number, and replace it. That means I haven't actually removed the same ball a third time. So here, it would simply be noted that we have 1 to the power of 64. Another person could say that since the pair I want is 1 - 1 - 1, it would mean that if we respect the principle, that is, I know that this is 4 to the power of 3, I want 1, that is to say, here, 1. How many is there since there's only one ball?

[00:16:10]That means it's one element, I want only one. Now, I want -1 - 1. So, I want two elements. 1 - 1.

[00:16:18]There are How many -1s are there? There's only one minus 1, and I want two elements.

[00:16:23]So it's simply 164. Is that clear for everyone?

[00:16:31]Good, Ariel, is that all right? Let's move on to the next question.

[00:16:54]Okay, I'm listening. What's the problem?

[00:17:02]You haven't understood at what level, man. At what level are you having difficulty?

[00:17:35]Okay, look, you have your moon, you have the one that contains four spheres. Don't forget that the principle is to say what? You have your one that contains exactly four spheres. And what do we want?

[00:17:48]We want m to be equal to A. And as you 've noticed, if I say that M is A, what does that mean? It simply means that if I want the coordinates of M, they are the same coordinates as those of A. That is to say, I want 1, I want -1, and I want it to be 1. Now, what is the probability I'm looking for? I know what? I simply know that since I want this, you're going to ask yourself the question. It's a bit like having something like this. You're asking yourself where to get a one. You have to choose now how many balls you actually have that have the number 1. There's actually only one ball. That's why I say you only have one chance of getting the number 1 because there are four balls and when you draw there, there's only one ball that has the number 1. So you only have one chance, that's why we say 1. Which means that to get a one, you only have one chance. There are four balls and when you draw, you only have one chance to get a one.

[00:18:49]Now you have to take a second ball that has the number -1. Or look Like the elements, how many are numbered? There's only one, and you want a ball. And then you want the number m₀-1.

[00:19:06]How many elements are numbered -1? There's only one element. The ball with the number t₀-1 is 1, and you want two elements. So, together you have a set of elements, and now you want a specific number.

[00:19:26]For example, here we said 4 to the power of 3, which means I have four balls, and I want three balls. So I want three elements. How many do I have? I have four. Now, as in my first question, I was told that M has the same coordinates as M. In other words, my first element should be 1, my second element should be -1, and my last element should be -1. Now, you're asking how many balls are numbered 1. There's only one element with the number 1. So, I raise 1 to the power of 1 because he wants only one ball with the number 1. Now, I have -1 - 1. That means I want two balls with the number mo- how many elements you have, four balls. There's only one ball with mo- take that twice. And if we don't have any order issues because we know I have to draw my first piece, it's 1 2nd - 1 3rd - 1. Is that right? We take one because that's the number of balls with the number 1.

[00:20:34][clears throat] There's only one. The number of balls with the number 1, there's only one. So I take the 1. Is that clear so far?

[00:20:45]Very good. Now, we're going to move on to The next question. In the next question, you are told this. You are told, let E1, that the event M belongs to the x-axis. Prove that the probability of E1 is equal to 1/4. Very well. I want to show that the probability of E1, what am I being told? Here, it's still about understanding probability when I say M belongs to the x-axis. If you belong to the x-axis, what does that mean?

[00:21:15]It means that your Y is 0 and your Z is 0. Your x-coordinate can only be 0, it can only be 1, it can only be -1? No, we don't have that problem. The essential thing is that your Y is 0 and your Z is 0 so that you can belong to the x-axis. So here, you are asked to show that this probability is 1 out of 4. So here, you Do the same thing. I know what? I was told that M belongs to the x-axis.

[00:21:44]If it belongs to the x-axis, that means its Y and Z coordinates don't exist. In other words, my point M would need to have an x-coordinate, regardless of its value. The essential thing is that my last two elements are zero.

[00:21:59]Is that clear so far?

[00:22:04]Please. Is that clear so far? You agree that for the probability of E1, what we're looking for, we're going to demonstrate that the probability of E1 is 1/4. But we already understand, I have a point M that belongs to the x-axis. If I belong to the x-axis, that means my Y is 0, my Z is 0. Now, my X doesn't matter, the essential thing is that Y is 0, Z is 0. Is that clear so far?

[00:22:34]Do we agree that this is Y0 Z0?

[00:22:42]Very good. Now the question would be, therefore, what will happen? What do we know? I simply know that here are my four balls.

[00:22:51]So here are my four balls. So I want to calculate this probability. We take, we want, three balls in total.

[00:22:59]So we keep our case, 4 to the power of 3.

[00:23:02]Now let's go. We know that x is anything. That means that the essential thing is that since x comes up once, the essential thing is that I'm going to take one of the four elements. That means that I can take x as -1. I can take x as 0. I can take x as 0, or I can take x as 1. That means that among the four elements, you're going to choose 1, and that's what will represent your probability; it's 4 to the power of 1.

[00:23:29]Now that Okay, that means the second element is 0. That means here, zero, how many zeos are there? There are two zeos. That means I'm going to say 2 to the power of 2 because I need 0.

[00:23:46]So look, look closely. The first element is x, it can be anything. It can be 0, it can only be 1, it can only be -1. The second and third elements are 0. Now, how many zeos are there among our balls? There are two zeos. That means I'm going to say I take from the number of zeos I have, it's 2, and I want 2 zeos. So the 0 is 0. The 2 that's here below represents the number of zeos you have in your bag.

[00:24:17]While the 2 of this The one up here actually represents the number of zeos you take to get your M. And as usual, like with the first question, if there's no other problem, it means we can't use a combination because we know exactly that from 0 comes the 2nd and 3rd positions; we know exactly where each one is placed. So we'll say 2 to the power of 2, so I'm left with 4 x 4.

[00:24:46]Now, we know that's 4. So we can conclude that this will give us 1 x 4. Am I following this proof? Is that clear?

[00:25:09]I'm listening, everyone. Is the proof clear?

[00:25:16]Are there any misunderstandings?

[00:25:21]Ariel, do you have any misunderstandings, Severine? Okay, let's move on to the next question.

[00:25:35]Very good. OK. Let's move on to the next question. Now, for the next question, you are asked P is a plane passing through O. O here is the origin of the coordinate system. That means that O has coordinates 0, 0.

[00:25:51]And we say a normal vector is simply a vector.

[00:25:59]Give a Cartesian equation of P.

[00:26:01]So, it's your turn, give me the Cartesian equation of P, that's easy.

[00:26:07]P is the plane that passes through O, the origin of our coordinate system, which therefore has coordinates 0, and a normal vector of P would simply be 1, 1, 1. Very good. A normal vector is indeed 1, 1, 1. I'm listening. Can someone give me the Cartesian equation of P? (disappears) So, for the writing, it would simply be to say let m be coordinates X, Y, Z belonging to P. Now, when you have this, we know that P has the equation AX + BY plus the C of Z + 1/D = 0.

[00:26:59]Good evening, F.

[00:27:02]Uh, I don't understand when you Okay, everyone, I want the Cartesian equation of P before we move on to how to write it up. You need to figure out how to find P.

[00:27:23]I want the equation of P.

[00:27:27]Back then, man, it's Verine. What does the equation of P give me?

[00:27:43]What does the equation of P give me?

[00:27:47]Ariel, we're going to find the equation of P. We said this. We were told that Y + Z = Z + D = 0. That's almost it.

[00:28:00]But you have to give the value of D. What is D? Don't forget, we gave you two pieces of information. We told you that P is the plane passing through O, where O is the origin of our P. So, it has coordinates 00. Now, its normal vector is N with coordinates 1, 1, 1, and we want the Cartesian equation of P.

[00:28:18]That's actually what we're asking for.

[00:28:23]4 to the power of x + y + z, that's all. You haven't forgotten anything, have you?

[00:28:39]Very good, x, y + z = 0, so... Here, when we have this, it means that here is your A, vo, and here is your C. So fa, and that's correct, it will give us X + Y + D, which equals 0. Now, you remember, we said that O belongs to P, that means we replace it here with the coordinates from above. So I have 0 + 0 + 0 + d, which equals 0. Thus, I have d, which equals 0, and as we said, the equation of our plane would simply be x + y + z = 0. Very good.

[00:29:34]Stop, I don't understand your question.

[00:29:35]When you say 4 to the power of 1, what does that mean? So, we have found the equation of P. Shall we move on to the last question? Is there a misunderstanding about this?

[00:29:49]The equation of P.

[00:29:53]So, really, that's good.

[00:29:58]Ariel.

[00:30:07]OK, we'll move on to the next question.

[00:30:09]If there are no questions, move on to the next one. If you're still confused, move on to the last question. Okay, here's an interesting question. You're told this, and it's even a point in the text. That means it's interesting. Very good, let's go. We'll move on to the next question. You're told this. We denote E2 as the event M belongs to the plane P, and we're asked what the probability of E2 is?

[00:30:36]Very good. Okay. What does E2 say? E2 simply says M belongs to the plane P. And what have we shown?

[00:30:48]We've shown that P has coordinates X + Y + Z = 0.

[00:30:57]Okay. The probability of 1 is K. Okay.

[00:31:04]You say no to this one. Very good.

[00:31:08]Look at what we were asked. We were told that M belongs to the x-axis. What does that mean? It means that its Y and Z are 0.

[00:31:17]The value of X doesn't matter. If I tell you that you started on the x-axis, that in fact You're not on the y-axis.

[00:31:24]Your y is 0 and your z is 0. That's actually what we said at this point. It means that the point is morally fixed for coordinates 0 here and 0 here, and on the x-axis, it can take any value. It can be -1, it can be 0, it can be 1.

[00:31:40]The essential thing is that Y and Z are equal to 0. So what we said here is simply that since X can take any value, what does that mean? How many possible choices do I have? I have 1, 2, 3, or 4 choices. So I'm going to take 4 to the power of 1 because there's only 1 x that you take from 4 to the power of 1. Now we multiply because we're not finished. We'll take two more numbers, or two more balls. Now the two balls that you have to Taking that must be the number zero. Now, how many elements do we have that have the number zero?

[00:32:15]We have two balls. Here I have two elements. And how many zeos do I want?

[00:32:19]I want zeos. Hence the 2 to the power of 2, which makes 4 x 4. We simplify the 4, we get one case, it doesn't work.

[00:32:42]So 4 to the power of 1, it's simply because x can take any of the four values ​​we have, 4 - 1, 0 or 1 doesn't matter, it takes any of the four values ​​you possess. Now you take zeos, how many balls have the number 2? There are two. And you take just how many elements? I take two elements. Very good. Let's go. So last question. So here, you are told I want M to belong to the plane. I know what? I know that when I am told that a point belongs to a plane, it means that it satisfies The equation of this plane. And since the equation of the plane is x + y + z = 0, that means, in other words, I need to find the different combinations so that by summing the three, we get zero. So here it will be finding the possible pairs to get zero. OK.

[00:33:40]Now, as for the possible pairs to get zero, we're going to try to... give me some of the combinations you want. We mustn't forget any pairs.

[00:33:48]I think that to get zero, we can have the pairs... We can take something as the first case we can have, which is that we choose... 0 3 times. That's for sure. If we take X 0 Y0 Z0, that's good. The second case you can have is that if you necessarily take... uh, if we take 1, we must necessarily take -1 and we must also take 0.

[00:34:21]And I think that's all we have.

[00:34:25]Very good. We can have... So you're saying this pair. Very good. So OK.

[00:34:35]This pair OK, both cases in general. So here, if I want zero, what does that mean? It simply means that for here, the number would be the probability, it would be that since I... OK.

[00:37:54]Very good.

[00:38:01]OK. Sorry, I have a technical issue.

[00:38:06]OK, let's go.

[00:38:14]And if we said we had to simplify everything, so there's the second case here. So first of all, we have to choose the position of 1 because in a little sorting, we need to know who is first, who is second, who is third. So here we have to choose the position of 1, of -1, and of 0. So in this case, it would be called a combination of 1 in 3, first because at the beginning, then we take the element 1. So here, since there's only one ball that has 1, it would be 1 squared. Apart from that, [clears throat] we continue. And the second ball, I chose the position of -1, but it There are only two positions left. And how many balls are there? There's only one ball, I'll say 1 to the quarter, sorry, one to the sale price. So OK, let's start from there. Let's say here for the second case, we know that to force 0, 1 + 1 equals 0. So we would need to have 1 - 1 and 0. You can allow this in different ways. But to go faster, it would simply be to choose.

[00:39:40]Start by saying I first choose the position of the first ball. We assume it's a number, since there are three positions, so it can be here, it can be here, or it can be here.

[00:39:48]So generally it's a combination of 1s in tr that will give tr, and now how many balls have the number 1? There's one ball and I just want one ball. So that's the power of 1.

[00:39:59]When you've finished choosing the position of your first ball and you just want one ball, you move on to the second ball. Now if I Let's take two positions. I'll choose the position of my second ball from the two and simply take it.

[00:40:17]How many balls have the number minus? There's only one ball and I only want one ball. And finally, there's only one place left. No need to choose the last place. But how many balls have the number zero? There are two and I only want one ball.

[00:40:31]So in other words, one approach would be... Simply going like this and saying that the probability we are looking for of E2 would simply be this. So 2 to the power of 3 + that is 3 x 1 uh x 2 x 1 x 2 is 4 x 4 x 4 so here 3 to the power of 3 we have 2 squared so I can put 2 squared factor of 10 I stay with 2 and if I stay with 2 3 and here it will give me 4 x 4 x 4 so which means I will simplify this so we have 5 it is 16 there you go what will give us the probability of e so that's for someone who starts directly now let's take the case of a student who actually wants to list all the possible cases. So, for example, all couples for 600 points. So I know that we can have Here are the different balls we have 0 1 now if so you get confused you make all the possible pairs, all the possible triplets. So I can have 0, I can have 1. So we're going to change the positions. You can have, we start with 1 - 1 0. You can have 1 0 - 1. You can have - 1 1 0, uh - 1 0 1, or 0 1 - 1, or 0 - 1 1.

[00:42:26]Here are actually all the pairs. There are 1 2 3 4 5 6 7 possible pairs. Now, what is the probability of getting 0?

[00:42:35]We have 0. So, how many elements do you want to know? 2, 3, and here would be first place, it's 1. There's only one ball with the number 1.

[00:42:49]Then a ball with the number -1 and two balls with the number 0. Same here, you have one ball with the number 1, you have two balls with the number 0 and you have one like that, you have 1 x 1 x 2 and if you have 1 x 2 x 1 and if you have 2 x 2 times how many?

[00:43:19]2 x 1 x [throat clearing] 1 and there you have 2 x 1 x 1. So that's repeating the same thing, which is 2 2 2 2 2 2, repeated six times, which means the probability will come to 2 to the power of 3 plus this, so + 2 x 6 is 4.

[00:43:53]Which means that's uh 2 x 2. We'll do 6, that's 2 x 3. So I can have 2 4, that's 12, so 4 x 3 and so on.

[00:44:05]Factoring 4 out, here I'm left with 2, always + 3 over 4 cubed, which makes uh 5 over 16. So there are two approaches: either someone doesn't directly say the large numbers together, it's this, or a sub- case. So we have seven replays. OK guys, that was the last question.

[00:44:29]Is there anyone who didn't understand that last question before we stopped? Don't forget, during these last two weeks try to have a fairly simple organization. Each thing is done at such and such a time in the morning, such and such a time in physics. You respect that, look at all the old topics. That's what will inevitably come back in another form, quite simply. And if you compose one that you would like us to work on together, simply write to me on WhatsApp.

[00:44:59]The number is this one or in the description, you will find the number. OK.

[00:45:06]Are there any misunderstandings regarding this last question? Herman Ariel Fa fawa, I hope he pronounces it correctly. Did you understand that last question?

[00:45:21]There are two approaches, only student of the This one is very good, please.

[00:45:32]OK, I think that's all for tonight and good evening everyone. The exercise is over.

[00:45:41]We will go by the grace of God. God willing, we will continue to do the live streams until the final exams, and when we finish with the Cameroonian baccalaureate, we will do live streams for other countries, Gabon, Comoros, and so on. Good evening everyone.

[00:46:02]Hm.