Okay, this is top 10 topic number far as expected value were on the business through 200 webpage will go to leaks and review review materials.  Statistics quick on statistics will click on statistics review.  Just a PowerPoint presentation it's only a few slides.  Click on okay, after following at home and will load PowerPoint load the statistics review and will scroll on down to topic number five to, which is located on slide in to view them in the second of is on the sly aid to 60 to see here/write this so will the better.  It is on slight 77, so do me a chance to find that slides like 77 expected value and will begin in a moment that expected value is an idea from probability about what value you're likely to get from to a especially if you if you'll draw particular value from a random brawl up and draw a particular member from random variable and up to you why it's important in statistics and it because expected value is also a weighted average.  It's a way to wait in various outcomes open the and for example is ski season is coming up in the Southern California area of the boss of the ski resort might one know how much revenue in my likely to make well how much revenue unlikely to make might be dependent on whether or not it snows a lot.  Its nose at average amount or whether or not its nose a little.  So the expectation of how much a ski resort operator makes might be related to the amount it smells for example or put differently, the expected value of of what the ski resort operator might make it depends on the probability of whether or not it's going to use snow a lot snow little or snow something in between.  So one thing and can and if all of the Cisco is no lots know little or snow something in between we could add up those probabilities say .5 .3 and .2 and those were bad to one of there was asked at the one million could multiply what we think we might make in terms of income times, each one of those probabilities in some the month that's what expected value is let me show you here on slight 78 he gives a definition expected value is a weighted average also along one average what we tend to see them along to pull him because my example of a ski resort is something we'd like to see in one year or but expected values generally use for long when target calculations watch because probabilities are not perfect that just estimates.  So for example, the formula there in the first bullet on on slide 78 is expected value is denoted as the use of acts.  The event is equal to the sum of all the ex-times to probabilities of all the axis to actually do some of the axis times a probability of axis and this is a and this is a expanded out aspects of one times the probability of ask someone a sex of two times a probability of one so this would be for example, in my ski resort example of this would be the how much money we can fix when it got to ask one is how much money we expect to make when it snows a lot piece of excellent one is the probability that snows a lot to say that it's .5 plus the how much we can expect to make if it snows a little say that $130,000 a year something like that times a probability that it might snow all a little say .2 so we multiply the amount that we can make when it snows a lot say 150,000 just making up that number hypothetical number of times a probability of the snows a lot .5 plus the probability must say 130,000 times a probability of .2 plus any other combinations of ex-values of how much we expect to make times to the probability of whether or not what's know as an example of a so so here's an example for a problem from the PowerPoint slides find the expected age at high school graduation is the Levin of the students were 17 sure sold 80 of the students were 18 years old and five of the students were 19 years old.  In the case of the first thing we have to do is add up to the number of students in each one of those three categories, 11 plus 80 plus five is 96 and then you'll see on the next page step two would make a small table for each one of those categories 17 and 18 and 19 with a week and that's asked we need to compute the probability of that so the probability is that number.  For example, there are 11 to 17-year-olds, the probability is a leaven divided by the total 96 Levin divided by 96 is .115 411 1/2 percent if you will.  Okay, so, what we got 11 because of the problem told us there were 1117-year-old slip a little of them and we had an 11 plus 80 plus five to get the whole number of students 96, and we just divide it with a way out we took we calculated the proportion of the number of 17-year-olds to the total.  So that's 11 divided by 96 is .114 .115 or 11.5%, which is to the same thing for the 18-year-olds 80 divided by 96 equals .83 trade we do the same thing for the 19 year olds.  There are five of them so five divided by 96 equals .052 and if we would add all those up.  We would see that the that number equals one or 100% equivalently and then what we do is we just take the number of individuals in the set that for every excuse me.  We take the age category, here are 17 we multiply times, the probability and of of those students which is .115 at the same thing as my probability of the amount of snows in the snowfall example to 17 times .115 is 1.995 and 18 times .3.833 is equal to 14.994.  19 is a 19 year old times .052, which is the probability of that will see some 19-year-old sets 5/96 as SQL 2.9 a day to and if we sum them all up and divide by three in other words, we take the average week at 17.937.  In other words, the expected value or the weighted average, what Bill of what the average ages of the students in this particular high school is 17.9 years all will do all we did was we waited the number of students by the probability we waited the ages scares me by the probability of the number students in a cage in there and mathematical_okay.  And that's it for expected value