Okay, So with this problem, we have a basan process with the rate of eight per hour. So that parameter is m equals 18. And the first part of this question is asking us to fund probability of six ships arriving in one hour on the fund probability of at least six, and then the probability of at least 10. So do that. We got our first part here, So we got the probability other ships arriving equals six. All right? And to do that Gap Probability six, eight Thetis, God, just plug it into the equation. Oh, yeah, E negative eighth. Sorry. I made a little clear you could be, uh, times eight to the sixth 86 Sorry. Um uh, there we go 8 to 6 over six. Sorry about that. My screen exactly were for a second, and then this answer comes out to be 0.1 to. So that's your answer to the first part of part of the next part. Asks us to find the probability of at least six ships arriving. Sort of do that. Find a probability that at least six we'll arrive. We've got one minus probability of less. Uh, West and six arriving. Alright. Right. And that makes sense because we can easily find a probability of less than six. But we will have no way of calculating the way of more than six taking this initial step. Right? Okay, to do this, they got one Linus summation, zero five a probability eight. Right for them. This equals, um, one minus summation E to the negative. Eighth times, eight to the X over X factorial. And this math comes out to be zero point. One of the hardest part about this is first making this initial step. That's a key step right there. And then you also just put numbers in the equation, right? This eight, in case you're wondering, comes from the, um, initial parameters that we were given. That is what? We're here in blue over here. Yeah. Look, we got em equals 80 against his t equals one. We got em equals eight. And that's where the eight that you're seeing that's coming from in all of these equations. All right, so that's your answer. The second part and then the last part of part pay possess, Mr Find a probability that a T least 10 ships arrive during this time, right? And we're gonna take the same step. It was enough for one should do one, minus the probability that West and 10 ships arrived for the same reason. Because we cannot possibly calculate the number between 10 and Infinity. So how? This is a very similar problem. One minus the summation. 0 to 9. Right here over nine. We don't include 10 because we're going less than, um, of, uh, okay. And ness equals one minus. Summation from 0 to 9. Off E to the negative beef times A to the X over X factor. Right. So if you look between this problem in the one in part B, the only thing that changes the parameters home of a summation, right, You're 509 Other than that, the same problem right here. Sorry about that. All right. And then this part. Sorry. This part comes out to 0.28 story. Okay. Okay. So that's party for you. Uh, let's go down. Well, tapping a part B heartbeats. Pretty self explanatory. Um, Harvey is actually find the expected value and standard deviation of ships arriving in a 90 minute time period. Right? Do this you still need. You need to use that same equation that we had written up there and blue. All right, again. Um, that equation is m equals eight c. And since you got a 90 minute window, Got 90 minutes times 60 minutes. Uh, sorry. 90 minutes. Time. One hour, over 60 minutes. Those units cancel, and here you get 1.5 hours case you're gonna put that in for tea, you get em equals Pete. 1.5 that you get. That's an important number for this part. Okay? Because the rest of this problem but early, the only thing to do is this equation right here. Why equals square root M, which equals a split of 12 which he calls 3.46 That is your answer for part B. So we don't think you have to do in that problem is apply the time that they give you into the promise they gave you. Take a square with that number and you got your answer. All right, so last part of this here, See, um, parte si is asking us to find the probability of 20 ships arriving in 2.5 hours. Tom and then probably of at most 10 arriving in 2.5 hours. So again, we have to start with that equation that they gave us. People ate tea, and they give us 2.5 hours. So this equals eight times two 0.5 in this equals 20. That's not what we're going to be using. And so we've got the probability 20 ships arriving in 2.5 hours and to do this year, we just do. Probability X is greater than 20. Matt equals one minus the probability that packs is less than 20 right? Samaj abuse before. Um, so this equation goes to one minus X summation 0 to 19. Um, probability of Pakistan 20. All right, we're going to use that same equation we did before in the problems above. And that looks like this. God. One minus the summation from 0 to 19 um, e to the negative 20th in times onesie to the axis over x factorial. That equals 0.5 three. Right. In case anybody's are confused. I wasn't very clear with this. The X that you're doing, it's It's these numbers here, right? X equals zero in 19. 0, I see, you're just adding, if you plug in zero for X here, you get that number. Plus the number you get from adding one in there, too. And they're always 90. You add all those up, and that is where you get here. 0.5 through Sorry for any confusion that I could have cleared up, uh, more transparent from start in. The last thing that we have to do for this problem, we'll find the probability that Oh, at most 10 ships come into the port in 2.5 hours. All right, so to do that, he used Probability X is less than or equal to 10. And this is easy. We're already looking for less than or equal to. We're less thing. Um, so this looks like the summation. X equals 02 And since it's since you got this or equal to sign, we've got you have these all those numbers summation of the probability of Okay, Z 2. 20. All right. In that look, something like this, the equation is going to be summation. X equals 0 to 20 of e to the negative 20th times once e to the x over X factorial I'm sorry. I'm sorry. That's not even 20. That's gonna be eating to the or the summation from zero town of even a 20 times. I went to the X over X factorial. That equals zero point 011 All right, that's your answer. I'm sorry for any confusion. That last part is well again. This you're doing it from zero to 10 not 25.