Five chips numbered 1-5 are placed in a jar. Two chips are drawn.
Let Z= the sum of the numbers.
WHat is the expected value of the random variable?
I know I have to use E = np
Can someone explain?
Add up the sum of the products of the sum and the probability of that sum.
Possible sums are 3,4,5,6,7,8 and 9.
There are 2 ways to get 3 or 9:
(1,2)(2,1); (4,5), (5,4)
There are 2 ways to get 4 or 8
There are 4 ways to get 5 or 7:
(1,4)(4,1)(2,3)(3,2); (2,5)(5,2);(4,3)(3,4)
There are 4 ways to get 6:
(1,5)(5,1)(2,4)(4,2)
The probability of getting 3,4,8 or 9 is 1/10. The probability of getting 5,6 or 7 is 1/5.
E = (1/10)(3+4+8+9) + (1/5)(5+6+7)
= 2.4 + 3.6 = 6
To find the expected value of a random variable, you are correct in using the formula E = np, where E represents the expected value, n represents the number of trials, and p represents the probability of a particular outcome.
In this case, the random variable is the sum of the numbers drawn from the chips. Let's go step by step to calculate the expected value:
1. Determine the probability of each possible outcome:
- There are five chips numbered 1-5, so the total number of possible outcomes is 5 * 5 = 25.
- Let's list all the possible sums when two chips are drawn:
- 1 + 1 = 2
- 1 + 2 = 3
- 1 + 3 = 4
- 1 + 4 = 5
- 1 + 5 = 6
- 2 + 1 = 3
- 2 + 2 = 4
- 2 + 3 = 5
- ...
(continue for all possible pairs)
- Count the number of possible outcomes for each sum. For example, there are two outcomes with a sum of 2: one outcome with (1, 1) and another with (2, 1). Calculate this for all possible sums.
2. Calculate the probability of each possible outcome:
- To find the probability of each outcome, divide the number of possible outcomes for each sum by the total number of possible outcomes (25 in this case).
3. Use the formula E = np:
- To calculate the expected value (E), multiply each sum by its corresponding probability.
- For example, if the sum of 2 has a probability of 0.08 and the sum of 3 has a probability of 0.12, you would multiply 2 by 0.08 and 3 by 0.12.
- Repeat this calculation for all possible sums and probabilities.
- Finally, sum up all the calculated products to find the expected value.
Note: In this case, since the probabilities are the same for each sum, which is 1/25, we can simplify the calculation by knowing that the expected value is the arithmetic mean of the numbers (1, 2, 3, 4, and 5), which is (1 + 2 + 3 + 4 + 5) / 5 = 15 / 5 = 3.