400 draws are made at random with replacement from 5 tickets that are marked -2, -1, 0, 1, and 2 respectively. Find the expected value of:
1. the number of times positive numbers appear
2. the sum of all the numbers drawn
3. the sum of the positive numbers drawn
the number of times positive numbers appear is 160
the sum of the positive numbers drawn is 240
Thanks you Din 123, the answer of B is 0 the 1 and 3 are correct.
To find the expected value of each scenario, we need to calculate the probability of each possible outcome and then multiply it by the corresponding value. Let's examine each case:
1. The number of times positive numbers appear:
The possible outcomes are 0, 1, 2, ..., 400. We need to calculate the probability of each outcome and multiply it by the value (number of positive numbers) for that outcome. Here's how we can do it:
- Probability of drawing a positive number (1 or 2): Since there are 5 tickets and 2 of them are positive, the probability of drawing a positive number is 2/5.
- Probability of drawing a non-positive number (-2, -1, 0): Since there are 5 tickets and 3 of them are non-positive, the probability of drawing a non-positive number is 3/5.
Now, let's calculate the expected value:
Expected value = (probability of drawing a positive number) * (value for positive number)
+ (probability of drawing a non-positive number) * (value for non-positive number)
Expected value = (2/5) * 1 + (3/5) * 0
= 2/5
Therefore, the expected value of the number of times positive numbers appear is 2/5.
2. The sum of all numbers drawn:
The value for each ticket is given as -2, -1, 0, 1, and 2. To find the expected value of the sum, we need to calculate the probability of each outcome (sum) and multiply it by the corresponding value. Here's what we can do:
- Probability of drawing -2: Since there are 5 tickets and only 1 is -2, the probability of drawing -2 is 1/5.
- Probability of drawing -1: Since there are 5 tickets and only 1 is -1, the probability of drawing -1 is also 1/5.
- Probability of drawing 0: Since there are 5 tickets and only 1 is 0, the probability of drawing 0 is 1/5.
- Probability of drawing 1: Since there are 5 tickets and only 1 is 1, the probability of drawing 1 is also 1/5.
- Probability of drawing 2: Since there are 5 tickets and only 1 is 2, the probability of drawing 2 is also 1/5.
Expected value = (probability of drawing -2) * (-2)
+ (probability of drawing -1) * (-1)
+ (probability of drawing 0) * (0)
+ (probability of drawing 1) * (1)
+ (probability of drawing 2) * (2)
Expected value = (1/5) * (-2) + (1/5) * (-1) + (1/5) * (0) + (1/5) * (1) + (1/5) * (2)
= -2/5 - 1/5 + 0/5 + 1/5 + 2/5
= 0
Therefore, the expected value of the sum of all numbers drawn is 0.
3. The sum of the positive numbers drawn:
To find the expected value of the sum of positive numbers, we need to calculate the probability of each outcome (sum of positive numbers) and multiply it by the corresponding value. Here's how we can do it:
- Probability of drawing 1: Since there are 5 tickets and only 1 is 1, the probability of drawing 1 is 1/5.
- Probability of drawing 2: Since there are 5 tickets and only 1 is 2, the probability of drawing 2 is also 1/5.
Expected value = (probability of drawing 1) * (1)
+ (probability of drawing 2) * (2)
Expected value = (1/5) * (1) + (1/5) * (2)
= 1/5 + 2/5
= 3/5
Therefore, the expected value of the sum of the positive numbers drawn is 3/5.