1.it has been reported that 83% of government employees has been migrated onto the single-spin salary structure.if a sample of 200 government employees is selected,find the mean variance and standard deviation of the number migrated onto the scheme.
2.if a fair die is tossed 12 times,find the probability of getting a 1,2,3,4,5 and 6 exactly twice each?
To find the mean, variance, and standard deviation of the number of government employees migrated onto the single-spin salary structure, we need to use the formula for these statistical measures.
Step 1: Calculate the mean:
The mean (μ) is the average value and is calculated by multiplying the probability of each outcome by the value of that outcome, and then summing them up. In this case, the possible outcomes are either an employee being migrated or not.
Given that 83% of government employees have been migrated, we can assume that the probability of an employee being migrated is 0.83 (or 83% written as a decimal). The probability of an employee not being migrated is the complement of the probability of being migrated, which is 1 - 0.83 = 0.17.
So, the expected value or mean (μ) is given by:
μ = (probability of being migrated) × (value of being migrated) + (probability of not being migrated) × (value of not being migrated)
μ = (0.83 × 1) + (0.17 × 0)
μ = 0.83
The mean number of government employees migrated onto the single-spin salary structure is 0.83.
Step 2: Calculate the variance:
The variance (σ^2) measures the spread or variability of the data. It is calculated as the sum of the squared differences between each value and the mean, divided by the total number of values.
Given that we have a sample of 200 government employees, and the probabilities of being migrated or not being migrated remain the same, we can use the variance formula for a binomial distribution:
σ^2 = n × (probability of being migrated) × (probability of not being migrated)
σ^2 = 200 × 0.83 × 0.17
σ^2 ≈ 28.22
The variance of the number of government employees migrated onto the single-spin salary structure is approximately 28.22.
Step 3: Calculate the standard deviation:
The standard deviation (σ) is the square root of the variance. It gives us a measure of how spread out the data is compared to the mean.
σ = √(variance)
σ = √(28.22)
σ ≈ 5.31
The standard deviation of the number of government employees migrated onto the single-spin salary structure is approximately 5.31.
Now, let's move on to the second question.
To find the probability of getting each number 1, 2, 3, 4, 5, and 6 exactly twice each when tossing a fair die 12 times, we will use the concept of combinations.
Step 1: Determine the number of ways to arrange the numbers 1, 2, 3, 4, 5, and 6 in pairs.
Since we need exactly two occurrences of each number, we can calculate the number of ways to arrange them.
6! / (2! × 2! × 2! × 2! × 2! × 2!)
= 720 / (2 × 2 × 2 × 2 × 2 × 2)
= 720 / 64
= 11,250
There are 11,250 ways to arrange the numbers 1, 2, 3, 4, 5, and 6 in pairs.
Step 2: Calculate the probability of each individual arrangement.
The probability of getting a 1, 2, 3, 4, 5, or 6 on a fair die is 1/6 for each number.
Step 3: Calculate the final probability.
Since we want exactly two occurrences of each number, we raise the probability of each number to the power of 2 (to account for two occurrences), and then multiply them together.
Probability = (1/6)^2 × (1/6)^2 × (1/6)^2 × (1/6)^2 × (1/6)^2 × (1/6)^2
Probability = (1/6)^12
Probability ≈ 0.000189
The probability of getting each number 1, 2, 3, 4, 5, and 6 exactly twice each when tossing a fair die 12 times is approximately 0.000189.