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Jamie's dad gave her a dye for her birthday she wanted to make sure it was fair so she took her dye to school and rolled it 500 times and kept track of how many times the dye rolled each number afterwards she calculated the expected value of the sum of 20 rolls to be 67 point 4 the expected value of the sum of 20 rolls to be 67 point 4 on her way home from school it was raining and two values were washed away from her data table find the two missing absolute frequencies from Jamie's data table so you see here she rolled her dye 500 times and she wrote down how many times she got a - she got a - 110 times a three 95 times a four seventy times a five seventy-five times and then she had written down which he got how many times she got a one and a six but then it got washed away so we need to figure out how many times she got a 1 and a 6 given the information on this table right over here and given the information that the expected value of the sum of 20 rolls is 67 point 4 so I encourage you to pause this video and think about it on your own before I give a go at it so first let's think about what what this this expected value the sum of 20 rolls being 67 point 4 tells us that means that the expected value of one roll the expected the expected value of the sum of 20 rolls is just 20 times the expected value of one roll so the expected value of a roll let me do it here expected value of a roll is going to be equal to 67 point 4 divided by 20 we can get our calculator out let's see so we have 67 0.4 divided by divided by 20 is three point three seven so this is equal to three point three seven so how does that help us well we know how to calculate an expected value given this this frequency table right over here if we say that this number right over here let's say that's capital A and let's say that this over here is capital B if we were to try to calculate the expected value of a roll what we really want to do is take the weighted frequency of each of these values the weighted sum so for example if we got a 1 a out of 500 times it would be a out of 500 times 1 times 1 plus plus I'll do these in different colors plus 110 out of 500 times 2 plus 110 out of 500 times 2 notice this is the frequency which was they got 2 times - we're taking a weighted we're taking a weighted sum of these values and then plus 95 out of 500 times 3 plus 95 out of 500 times 3 plus I think you see where this is going 70 over 500 times 4 plus 70 over 500 times 4 almost there plus let's see I haven't used this brown color plus 75 over 500 times I'll do it here plus 75 over 500 times 5 finally plus plus B over 500 plus B over 500 times times 6 this is going to give us our expected value of a roll which is going to be equal to three point three seven so all of this all of this is equal to is equal to three point three seven so one thing that we can do since we have all these 500s in this denominator right over here let's multiply both sides of this equation times 500 if we do that the left-hand side becomes well 500 times a over 500 is just going to be a plus 110 plus 100 and 110 times 2 so it's going to be 2 xx plus 95 tie beam's three that's going to be 15 less than 300 so it's going to be plus 285 plus 285 and then seventy times four is 280 plus 280 75 times five is going to be 350 plus 25 375 so plus 375 plus plus 6b plus 6b make sure I'm not skipping any steps here plus 6b is going to be equal to is going to be equal to this times 500 and that is going to be equal to three point three seven times 500 is equal to 16 85 1685 so all I did to go from this step right over here to K which I set up saying hey this is the expected value of one roll which is which we already know to be three point three seven is I just multiplied both sides of this equation by 500 I just did this times 500 and I did this times I did this times 500 and this 500 obviously cancels with all of these and then 500 times three point three seven is 1685 and so I got this right over here now I got one two three four five six yep I did I did enough I have a the right number of terms I just want to make sure I'm not making a careless mistake and so if we want to simplify this we can subtract two 2285 280 and 375 from both sides and so we would be if we did that we would get a if we subtract that from the left hand side we're just going to get a plus six be a plus 6b and on the right hand side we are going to get let's get our calculator out 1685 minus minus 220 220 minus 285 - 285 minus 280 minus 280 - 375 - 375 gets us to 500 five so we get a plus 6b is equal to 525 and you say okay you did all that work but we still have one equation with two unknowns how do we figure out what a and B how do we figure out what a and B actually are well we know something else we know and this was actually much easier to figure out we know that the sum of this whole table right over here a plus 110 plus 95 plus seventy plus 75 plus B is equal to 500 or if we let me write that down so we know we know that a plus plus 110 plus 95 plus seventy plus 75 plus B plus B needs to be equal to 500 needs to be equal to 500 or we could subtract 110 plus 95 plus seventy plus 75 from both sides and get I should subtract it from the left-hand side you're just left with a plus B a plus B and on the right hand side if we start with we start with 500 so 500 minus 110 minus 95 minus 70-75 gets us to 150 so a plus B must be equal to 150 is equal to 150 and now we have a system of two equations and two unknowns and so we know how to solve those we could do it by substitution or we could subtract the second equation from the first so let's let's do that let's subtract the left-hand side of this equation from that so or essentially we could add these two can multiply this 1 times a negative 1 and then add these two equations the A's are going to cancel out and we are going to be left with 6 B minus B is 5 B is equal to 375 is equal to 375 did I do that right if I had 125 to this I get to 500 then another 25 I get to 25 so 5b is equal to 375 or if we divide both sides by 5 we get B is equal to B is equal to 75 B is equal to 75 so this right over here is equal to 75 if B is equal to 75 what is a well we know that a plus B is equal to 500 we figure that out a little while ago before we multiply both sides of this times a negative 1 we knew that a plus B a plus B when B is now 75 so we could say a plus 75 is equal to is equal to 150 is equal to 150 and that's just from from this we figured out that a plus B is equal to 150 before we multiply both sides times a negative subtract 75 from both sides you get a is also equal to 75 and we are done