Suppose I 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 I have to add before the expected value is at least 3.5?
Let k=number of 4's to add
then
E(x)=Σx*P(x)
=(8*2+(2+k)*4)/(8+2+k)
If E(x)≥3.5
then
(8*2+(2+k)*4)/(8+2+k)≥3.5
solve by cross multiplication
16+8+4k≥35+3.5k
0.5k≥11
k≥22
To determine the number of 4's you need to add to the bag before the expected value is at least 3.5, we need to calculate the current expected value and then iteratively add 4's until the expected value reaches or exceeds 3.5.
Let's start by calculating the expected value with the current set of slips in the bag:
1. Calculate the probability of drawing a slip with a 2: There are 10 slips in total, and 8 of them have a 2 on them. Therefore, the probability of drawing a slip with a 2 is 8/10 or 0.8.
2. Calculate the probability of drawing a slip with a 4: There are 10 slips in total, and 2 of them have a 4 on them. Therefore, the probability of drawing a slip with a 4 is 2/10 or 0.2.
3. Calculate the expected value: Since each slip with a 2 has a value of 2 and each slip with a 4 has a value of 4, the expected value of a single draw can be calculated as follows:
Expected value = (Probability of drawing a 2 * Value of a 2) + (Probability of drawing a 4 * Value of a 4)
= (0.8 * 2) + (0.2 * 4)
= 1.6 + 0.8
= 2.4
The current expected value is 2.4, which is below 3.5.
To increase the expected value to at least 3.5, we can add more slips with a value of 4 to the bag until the relative contribution of the slips with a value of 4 is high enough.
4. Determine the target expected value: Subtract the current expected value from the desired expected value:
Target expected value = 3.5 - 2.4
= 1.1
5. Calculate the number of 4's needed: Since each slip with a value of 4 contributes 1.1 to the expected value, divide the target expected value by 4:
Number of 4's needed = Target expected value / Value of a 4
= 1.1 / 4
≈ 0.28 (rounded up to the nearest whole number)
Therefore, you need to add at least 1 additional slip with a value of 4 to the bag to make the expected value at least 3.5.
To determine how many 4's you need to add before the expected value is at least 3.5, we can use the formula for expected value (also known as mean or average).
The expected value (E) is calculated by multiplying each possible outcome by its probability and summing them all up.
Let's break down the steps:
1. Initially, you have 10 slips of paper in the bag. Eight slips have a 2 on them and two slips have a 4 on them. So we have the following probabilities:
- Probability of drawing a 2: 8/10 = 0.8
- Probability of drawing a 4: 2/10 = 0.2
2. The expected value (E) can be calculated as follows:
E = (2 * probability of drawing a 2) + (4 * probability of drawing a 4)
= (2 * 0.8) + (4 * 0.2)
= 1.6 + 0.8
= 2.4
3. Since the initial expected value is 2.4, we need to add more 4's to increase it to at least 3.5.
4. Let's say we add x more 4's to the bag. Now, the total number of slips in the bag would be 10 + x. The probability of drawing a 2 would remain the same (8/(10 + x)), and the probability of drawing a 4 would be (2 + x)/(10 + x).
5. We need to set up an equation to solve for x when the expected value is at least 3.5:
(2 * 0.8) + ((2 + x) * (0.2/(10 + x))) ≥ 3.5
6. Simplifying the equation:
1.6 + (0.4 + 0.2x)/(10 + x) ≥ 3.5
7. Cross-multiplying and rearranging the equation:
1.6(10 + x) + 0.4 + 0.2x ≥ 3.5(10 + x)
16 + 1.6x + 0.4 + 0.2x ≥ 35 + 3.5x
8. Combining like terms:
2.0x + 16.4 ≥ 35 + 3.5x
9. Subtracting 2.0x from both sides:
16.4 ≥ 35 + 1.5x
10. Subtracting 35 from both sides:
-18.6 ≥ 1.5x
11. Dividing by 1.5:
-12.4 ≥ x
12. Since x represents the number of additional 4's we need to add, it must be a positive integer. Therefore, the smallest integer greater than or equal to -12.4 is -12. So, you need to add at least 12 additional 4's to the bag before the expected value is at least 3.5.