+42 a.) What is the expected value of the number shown when we draw a single slip of paper?
b.) What is the expected value of the number shown if we add one additional 4 to the bag?
c.) What is the expected value of the number shown if we add two additional 4's (instead of just one) to the bag?
d.) How many 4's do we have to add to make the expected value equal to 3?
e.) How many 4's do I have to add before the expected value is at least 3.5?
+23252 8 slips of paper each has a 2 on them; 2 slips of paper each has a 4 of them
The total value of all the slips when added together is (8)(2) + (2)(4) = 16 + 8 = 24
The expected value of one draw is the average value of these 10 slips of paper is 24 / 10 = 2.4
If we add one additional 4, then the total becomes 24 + 4 = 28.
Since there are now 11 slips of paper, the expected value (average) is 28 / 11 = 2.5454...
Adding another 4, the total becomes 28 + 4 = 32.
Since there are now 12 slips of paper, the expected value (average) is 32 / 12 = 2.6666...
How many 4s must be added to get an average of 3?
Starting with the original 10 slips of paper and the original sum of 24:
let n represent the number of slips of paper (each containing a 4) that we have to add to the original 10.
The sum then becomes: 24 + 4n
The number of slips of paper becomes 10 + n.
The average becomes the sum divided by the number of slips of paper: (24 + 4n) / (10 + n).
Since we want the average to be 3, we can create this equation: (24 + 4n) / (10 + n) = 3
Solving this by multiplying both sides by (10 + n) ---> 24 + 4n = 3(10 + n)
---> 24 + 4n = 30 + 3n
---> n = 6
We would have to add 6 slips to the original 10.
How many 4s must be added to get an average of at least 3.5.
Using the technique of the previous problem, we get the equation: (24 + 4n) / (10 + n) > 3.5
Can you take it from here?
