In problem three were given a distribution of values 77-81. And the probability for each of those values Question a asks us to determine the probability that we would get exactly 80 and that's going to be just reading it straight from the table. The probability of 80 is 0.40 as we can see right there. Problem be asks for the probability of a value greater than 80. Now. It's important to take note to the fact that we are just talking about greater than 80 and not 80. Exactly. So the only value that's greater than 80 is 81 and the probability of getting an 81 is 0.10 part C asks for the probability of At most 80 being less than or equal to 80. So that would be from 80 down. Now there's kind of two ways that you can go about this. Um you could add up before values 77, 78, and 80 those four probabilities or really the only thing that we're excluding is the 81. That's the only value that does not satisfy this. So we don't want the 10%. Instead we want The 90% or 0.90. And again you could have added the .15.15 would be .3 Plus .2 would be .5 and then plus .4 would be .90. Now D. E and F. We're going to require a bit more work. We want to determine the mean of the probability distribution. So this is very similar to the idea of doing a weighted average. And so for part D the mean of this distribution is going to be the sum of each of the possible values Multiplied by its probability. So 77 times .15 Plus, times .15 plus, 79 Times .20 Plus. And I'm gonna jump down the line here 80 Times .40 plus 81 Times .10. And so I'm just going to take a moment and plug these into the calculator. And so 77 times 770.15 plus 78 times 780.15 plus, 79 times 790.20 plus, 80 times .4 plus 81 times .1. And I'm sorry, I think you have to do that again. I didn't did not get the correct answer, but I can see what I did wrong, 79.15 should be the correct answer. For part E we want the variants of the probability distribution. And then for part f we're going to convert that to the standard deviation. So, for variants you need to um take and and this is very similar to your variance formula. Just when you're dealing with a data set, we're going to take each possible value and subtract the mean square that and multiply it by its probability. So again, it's kind of like doing the idea of a weighted average. So this is going to be for the variants sigma squared sub x X, just representing the probability distribution that we're talking about, We have a possible value of 77 minus the mean squared, multiplied by that weight 0.15 I'm going to move this over a little bit, so I have some more space here, then we add to that 78 -79.15 Squared, multiplied by .15 plus 80 -79.15 Squared. And that is, I'm sorry, I missed the 79. Let me get that in there. So 79 -79.15, multiplied by .20 Plus. Now the 80 -79.15 Squared, And it's probability is .40. And then lastly, 81 -79.15 squared, Multiplied by .10. So then you carefully enter this into the calculator. It does help if you're able to do a little bit of mental math. So, for instance, I just knew that 77 -79.15 would be negative 2.15. So I entered that into my calculator squared Times .15. Just make sure that you put the negative 2.15 in parentheses with the square on the outside, or don't even worry about the negative. That can work too. Okay. Yeah, yeah, So I have an answer for the variants of 1.5275, again, just entering everything that we have there into the calculator. Now, the nice thing for part F, we want the standard deviation for the probability distribution, we're just going to take the square root of the variance, And that gives you approximately 1.236, Yeah.