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?
To find the expected value, we need to multiply each possible outcome by its probability and then sum them up.
Given the original bag with 10 slips:
- There are 8 slips with a 2, and the probability of drawing a 2 is 8/10 = 4/5.
- There are 2 slips with a 4, and the probability of drawing a 4 is 2/10 = 1/5.
If we add two additional 4's, the total number of slips in the bag becomes 12 (10 original slips + 2 additional slips).
The expected value can be calculated as follows:
Expected value = (Probability of drawing a 2 * Value of 2) + (Probability of drawing a 4 * Value of 4)
= ((4/5) * 2) + ((1/5) * 4)
= (8/5) + (4/5)
= 12/5
= 2.4
Therefore, the expected value of the number shown is 2.4.
To find the expected value, we need to calculate the sum of all the possible outcomes weighted by their probabilities.
In this case, we have 10 slips of paper in the bag. Before adding the additional two 4's, there are 8 slips with a 2 and 2 slips with a 4. Therefore, the possible outcomes and their probabilities are as follows:
- Probability of getting a 2: 8/10 = 0.8
- Probability of getting a 4: 2/10 = 0.2
Now, let's consider what happens when we add the two additional 4's. The new possible outcomes and their probabilities are:
- Probability of getting a 2: 8/12 = 0.67
- Probability of getting a 4: 4/12 = 0.33
To calculate the expected value, multiply each outcome by its probability and sum them up:
Expected value = (0.67 * 2) + (0.33 * 4) = 1.34 + 1.32 = 2.66
Therefore, the expected value of the number shown is 2.66 when we add two additional 4's to the bag.