Were given that delightful has a 0.2 chance of failing, and each box has ate light bulbs in it. So the first thing we're gonna do is find the probability distribution. Now, on this page, on the last page of the book, you were given a lot of notes about how to do these problems on a computer. That's because they want you to do these problems on either a computer or graphing calculator. I'm gonna be doing this and excel well, you conduce this in over programs. The steps might be a little bit different, though, So if we do this in Excel, first thing I'm gonna do is make a list of all the possible outcomes. There's eight trials, so we need the number 0 to 8. Just this select all home and dragged down. They don't want to going to be one under formulas and statistical, and I use the binomial distribution. Tual. So our trials that's eight. Cause there's ate light bulbs in the box or probability was 80.2 We don't have to convert it like we normally deal because we weren't given it as a percentage. It's already in the form we wanted to be in Cumulative is false for not working on this right now. So for this the number I'm gonna play it all my numbers from 0 to 8. So starting with B a one box colon, the A nine rocks, and we get this. So now we need to draw a hist. A gram of acsa. So for a history Graham of acts, we put everything in order, and our X values are going to be the X access. So that's just gonna be the numbers from 0 to 8. So to free 456 seven and eight. No wonder why access is gonna be all of the probabilities. However, what? Let's go back to take a look here. You notice the 1st 1 is the biggest. Then the 2nd 1 smaller, smaller, smaller and keeps getting smaller and smaller. This e means times 10 to the blank power. So, for example, on the probability of seven bull to being bad is one times 10 to the negative, a lower power. So the reason why it keeps getting smaller is because of this probability is so low that getting zero, where wine that's the most realistic outcome so we can make our hissed a gram like a descending staircase. Kind of. So zero is gonna be the highest, then wine and to than free four and eventually these ones, they're going to get so small that it's just a tiny little dash. And then we'll fill in some values with one for zero. That was a point. Eat by one one at one that was 10.139 Then we have points cereal 099 I don't put one more. The one at free. We have 10.0 for. So this is Ah, a decent history, Graham. So what is the probability of getting a box with no failing light bulbs? So let's go back to our distribution that we just made it. So no feeling like bulbs that zero. So that's right here. The 0.851 This problem is just reading the table. So 0.851 That's because the no here means zero. What is the probability of getting a box with no more than one? So what we'll do for this one is will take all the possible options. No more than one means even zero or one. So we'll take both of these two and add them together. And when you do that, you should get a point 990 We just add the 1st 2 together, find the mean and standard deviation of X. So because we have a binomial, random variable over the light bulb fails or it works. So it's binomial. These are the two equations we used to get mean and standard deviation. Your notes might say a que here instead of one minus p. That's the same thing. So N is our number of trials, that is eight and P is our probability. That's 80.2 so are mean is equal. Chill eight times 80.0 chill, which is equal 2.16 Our standard deviation is the square root of eight times 80.0 chill times one minus 10.2 or 0.98 If you're using keel, it's the same feign, and this gives us point free 96 So what proportion of the distribution is between you have these two symbols. So first, let's define them. This one is called new, and it's the mean. This one is called Sigma, and it's the standard deviation. So this is asking how much of the data is between mean minus standard deviation and mean plus standard deviation. Well, mean minus dinner deviation. That gives us a negative. Zero is our lowest option. So we have starting from zero chill. But what's 0.16 plus 0.396 Find six. Well, that's Ah, 0.5 by six. So we go here if you notice we have all whole numbers here. So 0.556 That doesn't get us toe one because this is binomial. We can't have half of a try algo ever way. We can't have a light bulb half working, so we have to round everything down and we're only gonna be in the zero box, for example. Let's say we got a mu plus Sigma was equal to free. Then we would include zero to free, but because it hasn't hit one, we're going to stay in the zero box. So we're gonna get 0.851 again. Now, what about if we multiply the standard deviation by two first? So we're still gonna have zero here? It doesn't matter what the exact value is its negative. So we're still gonna have zero, and we're gonna get 00.952 here again. It didn't hit one yet, so we're still within that 85% of data or 0.851 So how does this information relate to the empirical rule and Chevy sex? Fear out. These two rules give less guidelines for how much of the data is going to be within one standard deviation of the mean, like in part E two standard deviations of the mean in free standard deviations of the mean. And sure enough, most of the data should be off in one standard deviation last living too and less within free. However, in our problem, because the probability of a light bulb not working is so low. Almost all of the data is within one standard deviation and so and relates to this because this is what we would expect from such a low probability. Okay, if we buy 100 box isn't we're going to simulate how many of them were go bad. There are many different ways of doing random numbers simulations. The way I'm going to do it is just one way of doing it. So what? I'm gonna do is generate ah 100 random numbers ranging from 1 to 100. I'll do this by going to insert function under category all, and I'm going to scroll down to the ours and almost the aerial ran between. I'm gonna generate numbers from 1 to 100. Now, what my role is going to be is any time we see the number one or two, that means it is going to be a broken bull. That's because we have a 20.2 chance of a light bulb failing. So that's really 2%. So there's 100 boxes, 2%. So wanting to that's 2% of 100 because that's two numbers. Now, we're going to do this 100 times, so I'm going to drag this all the way down to Ah 100. I'm gonna quickly look through and see if we got any ones or two. I see it too. I see one. Okay, so we got to Obviously, your answer could realistically be 0123 or four. Now, repeat, part h in common on it. So we're gonna do this a few times now. We're gonna comment on what we see happening, Nina. So it's high, like this whole thing. This is why I'm doing it this way. So is gonna be very easy to recreate, because I just drag this to the right. I'll do free more. Okay, so starting at the 2nd 1 I see a to there's a one. A number one so free. So we got to for the 1st 1 Been free. Now let's look at this line. No. Is that looks like zero must quickly make sure Yep, zero. And then the last one. There's a one. There's an over one. So, too, to so free we got free. This time you notice we're kind of getting around the same amount each time to or free makes sense. There's a 2% chance that a bold goes bad, so the higher this number gets, we should be getting closer to 2% each time. So 12 or free