The arrival times of the bus Alex takes to work follow a normal distribution, with μ = 12 min \mu=12\min after the scheduled arrival rime & σ = 3 min \sigma=3\min . If the bus is scheduled to arrive at Alex's work 10 min before opening, what is the probability that Alex arrives on time (i.e. the bus is less than 10 min late)?

Now that we've seen how to find the probabilities with nonstandard normal distributions, let's take a look at how this works in the real world and look at this practice problem. The wording's a little bit tricky with the minutes, but we're gonna go ahead and break it down together. Alright? Now the arrival times of a bus that this person named Alex takes to work follows a normal distribution. So in other words, it's sort of a bell shaped curve, kind of like this. I like to just draw a picture here. Now what we have here is that the mean time that it takes, mu, is equal to twelve minutes after the scheduled arrival time. In other words, this bus tends to run a little bit late here. And the standard deviation is equal to three minutes. So basically, everything works just like a normal model would, except instead of a mu being one, a standard deviation of or sorry. Mu is zero and standard deviation of one, we have a mu equals 12 and a standard deviation, sort of how wide the data is, is equal to three minutes. Alright? Now if the bus is scheduled to arrive at Alex's work ten minutes before it opens, what's the probability that Alex actually arrives on time? Remember, this bus tends to run late, and it tends to run an average of twelve minutes sort of late. So in other words, what they're really sort of asking us here is what's the probability that the bus is less than ten minutes late? Okay? So the bus on average is twelve minutes late, but we're sort of trying to figure out what's the probability that it's less than 10. So twelve would be over here somewhere in the middle, and ten would be somewhere over here. So basically, what we're trying to find here is our x value is equal to 10. What is the probability or the area, of sort of all of the arrivals that are less than ten minutes late, which means that Alex would actually make it to work on time? So this is basically what we're trying to find over here. This is gonna be the percent, that Alex is on time. Alright? That's basically what we're trying to find here. So we can go ahead and take all of these numbers, and we can transform them back into a standard normal distribution problem by taking a look at our z score equation. We have a mu is equal to 12, our sigma is equal to three, our specific x value that we're interested in is 10. So basically, what we're trying to find here is what's the probability that x is less than 10. We can transform this into a z score using our brand new equation that we've seen over here, which is really just that 10 minus 12 divided by three. In other words, what's the x value minus the mean? How far is that away relative to the standard deviation? What you should go ahead and find here is this is equal to negative 0.67. It's gonna be a decimal, but you would just round it like this. So in other words, the probability that X is less than ten minutes late is the same exact thing as saying what is the probability that in a standard Z distribution or normal distribution, you would have a Z score of less than negative 0.67. It's the exact same problem. Okay, so you can use a z table to look this up or you can use some technology. Let's just go ahead and take a look at our z table over here. We would look at the negative values. We would go find where negative 0.67 is, which is going to be this row right over here where it intersects with this row over here where it intersects with this column down here. And we would find that it's basically this number which is negative or 0.2514. Alright? So let's go back to our problem over here. Basically, the probability that x is less than 10 would be, again, this probability over here, which we've seen how to do. The calculator would tell you the same exact thing, and this is just going to be 0.2514. We can actually just go ahead and express this as a percent by shifting the decimal place, and this would be 25.14%. In other words, the probability that Alex actually makes it on time is about 25% or one in four. So probably not a great gamble to make there considering this bus tends to run a little bit late. Alright? So let's go ahead and take a look at some more practice problems. Thanks for watching.