Hi! I need urgent help for these questions here. They have a deadline for today and I have no idea how to solve them
(a). Suppose we have a bag with 10 slips of paper in it. Eight of these have a 2 on them and the other two have a 4 on them.
What is the expected value of the number shown when we draw a single slip of paper?
(b). Suppose we have a bag with 10 slips of paper in it. Eight of these have a 2 on them and the other two have a 4 on them.
What is the expected value of the number shown if we add one additional 4 to the bag?
(c). Suppose we have a bag with 10 slips of paper in it. Eight of these have a 2 on them and the other two have a 4 on them.
What is the expected value of the number shown if we add two additional 4's (instead of just one) to the bag?
(d). Suppose we have a bag with 10 slips of paper in it. Eight of these have a 2 on them and the other two have a 4 on them.
How many 4's do we have to add to make the expected value equal to 3?
Please remember to get to these questions! I am in an EMERGENCY situation.
Dude, just enroll in a easier class. You are literally asking to for people to do problems for you. Im assuming you use AoPS but if its a different website, just press the give up button. You need to actually try, not ask everyone else to do it for you.
These are very interesting problems.
If you show some efforts in solving them, there would be likely to be more chances of response.
Fricking cheated.
Yeah its not that hard, because you obviously can't solve any of theese!
I am working on this too. Just read your transcript. That's what I did and it helped a lot. DON'T BE LAZY!
I understand that you need urgent help with these questions. Let's go through each question one by one.
(a) To find the expected value of the number shown when we draw a single slip of paper, we need to calculate the average value.
In this case, we have 8 slips with a 2 and 2 slips with a 4.
To calculate the expected value, we multiply the value of each slip by its probability and sum them up.
Expected value = (2 * Probability of drawing a 2) + (4 * Probability of drawing a 4)
Probability of drawing a 2 = Number of slips with 2 / Total number of slips = 8 / 10 = 0.8
Probability of drawing a 4 = Number of slips with 4 / Total number of slips = 2 / 10 = 0.2
Expected value = (2 * 0.8) + (4 * 0.2) = 3.2 + 0.8 = 4
Therefore, the expected value when drawing a single slip of paper is 4.
(b) Now, let's consider the scenario where we add one additional 4 to the bag. The total number of slips in the bag is now 11 (10 original slips + 1 additional slip).
We still have 8 slips with a 2 and 3 slips with a 4 (2 original slips + 1 additional slip).
Following the same steps as in part (a), we calculate the expected value:
Probability of drawing a 2 = 8 / 11 ≈ 0.727
Probability of drawing a 4 = 3 / 11 ≈ 0.273
Expected value = (2 * 0.727) + (4 * 0.273) ≈ 1.454 + 1.092 ≈ 2.546
Therefore, the expected value when drawing a slip of paper after adding one additional 4 is approximately 2.546.
(c) Now, let's consider the scenario where we add two additional 4's to the bag. The total number of slips in the bag is now 12 (10 original slips + 2 additional slips).
We still have 8 slips with a 2 and 4 slips with a 4 (2 original slips + 2 additional slips).
Following the same steps as before, we calculate the expected value:
Probability of drawing a 2 = 8 / 12 ≈ 0.667
Probability of drawing a 4 = 4 / 12 ≈ 0.333
Expected value = (2 * 0.667) + (4 * 0.333) ≈ 1.334 + 1.332 ≈ 2.666
Therefore, the expected value when drawing a slip of paper after adding two additional 4's is approximately 2.666.
(d) To find out how many 4's we need to add to make the expected value equal to 3, we can set up the equation:
(2 * Probability of drawing a 2) + (4 * Probability of drawing a 4) = 3
Using the given information from part (a), we have:
(2 * 0.8) + (4 * 0.2) = 3
1.6 + 0.8 = 3
2.4 = 3
To make the expected value equal to 3, we need to add enough 4's until the probability of drawing a 4 is 1. So, we would need to add 8 more slips with a 4, bringing the total number of 4's to 10.
I hope this helps you with your emergency situation. If you have any further questions, please let me know.