1.find the mean,variance and standard deviation for the number of heads when 10 coins are tossed.
2.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.
3.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 for each scenario, we need to use some basic formulas and concepts from probability and statistics.
1. Finding mean, variance, and standard deviation for the number of heads when 10 coins are tossed:
a) Mean: The mean is calculated by multiplying the probability of each possible outcome by the corresponding value and summing them up. In this case, the possible outcomes range from 0 to 10 heads, so the mean is:
Mean = (0 x P(0 heads)) + (1 x P(1 head)) + (2 x P(2 heads)) + ... + (10 x P(10 heads))
b) Variance: The variance is a measure of how spread out the data is. It is calculated by finding the average of the squared differences between each data point and the mean. In this case, since the possible outcomes range from 0 to 10 heads, the variance can be calculated using the formula:
Variance = (0^2 x P(0 heads)) + (1^2 x P(1 head)) + (2^2 x P(2 heads)) + ... + (10^2 x P(10 heads)) - Mean^2
c) Standard Deviation: The standard deviation is the square root of the variance and helps in understanding the typical dispersion of the data around the mean. Thus, in this case:
Standard Deviation = Square root of Variance
2. Finding the mean, variance, and standard deviation for the number of government employees migrated onto the single-spin salary structure:
a) Mean: Since 83% of government employees have migrated, we can calculate the mean by multiplying the probability of each possible outcome by the corresponding value and summing them up. In this case, the possible outcomes range from 0 to 200 (since there are 200 government employees in the sample), so the mean is:
Mean = (0 x P(0 migrated)) + (1 x P(1 migrated)) + (2 x P(2 migrated)) + ... + (200 x P(200 migrated))
b) Variance: Using the same logic as in the previous question, we can calculate the variance using the formula:
Variance = (0^2 x P(0 migrated)) + (1^2 x P(1 migrated)) + (2^2 x P(2 migrated)) + ... + (200^2 x P(200 migrated)) - Mean^2
c) Standard Deviation: Similarly, the standard deviation would be the square root of the variance.
3. Finding the probability of getting each outcome exactly twice in 12 tosses of a fair die:
For each number (1, 2, 3, 4, 5, 6), the probability of getting it exactly twice can be calculated using the binomial probability formula:
P(X) = (n choose x) * p^x * (1-p)^(n-x)
In this case, n = 12 (since the die is tossed 12 times), x = 2 (since we want to get each number exactly twice), and p = 1/6 (since there is an equal probability of getting any of the six numbers on each toss).
You can calculate P(X) for each number using the above formula, and the sum of these individual probabilities would give you the overall probability of getting exactly two of each number in 12 tosses.
1. To find the mean, variance, and standard deviation for the number of heads when 10 coins are tossed:
Step 1: Determine the possible outcomes. When a coin is tossed, there are two possible outcomes - heads or tails. Therefore, the total number of possible outcomes when 10 coins are tossed is 2^10 = 1024.
Step 2: Calculate the mean.
The mean (or expected value) is given by:
mean = (number of heads/total possible outcomes) * 10
Since each coin toss has a 0.5 probability of getting heads or tails, the mean can be calculated as follows:
mean = (0.5 * 10) = 5
Therefore, the mean number of heads when 10 coins are tossed is 5.
Step 3: Calculate the variance.
The variance is given by:
variance = (number of heads - mean)^2 / total possible outcomes
Since the variance is the same for both heads and tails, we can calculate it using the number of heads:
variance = (5 - mean)^2 / total possible outcomes
variance = (5 - 5)^2 / 1024
variance = 0
Therefore, the variance for the number of heads when 10 coins are tossed is 0.
Step 4: Calculate the standard deviation.
The standard deviation is the square root of the variance.
standard deviation = sqrt(variance)
standard deviation = sqrt(0)
standard deviation = 0
Therefore, the standard deviation for the number of heads when 10 coins are tossed is 0.
2. To find the mean, variance, and standard deviation of the number migrated onto the single-spin salary structure:
Step 1: Calculate the mean.
The mean is given by:
mean = (percentage migrated/100) * sample size
Therefore, the mean can be calculated as follows:
mean = (83/100) * 200
mean = 166
Therefore, the mean for the number migrated onto the scheme is 166.
Step 2: Calculate the variance.
The variance is given by:
variance = (percentage migrated/100) * (1 - (percentage migrated/100)) * sample size
Therefore, the variance can be calculated as follows:
variance = (83/100) * (1 - (83/100)) * 200
variance = (83/100) * (17/100) * 200
variance = 28.22
Therefore, the variance for the number migrated onto the scheme is 28.22.
Step 3: Calculate the standard deviation.
The standard deviation is the square root of the variance.
standard deviation = sqrt(variance)
standard deviation = sqrt(28.22)
standard deviation = 5.31
Therefore, the standard deviation for the number migrated onto the scheme is 5.31.
3. To find the probability of getting each number on a fair die exactly twice when tossed 12 times:
Step 1: Determine the total number of possible outcomes. When a fair die is tossed, there are six possible outcomes - 1, 2, 3, 4, 5, or 6. Therefore, the total number of possible outcomes when tossed 12 times is 6^12 = 2,176,782,336.
Step 2: Calculate the possible ways to get each number exactly twice.
We need to calculate the number of ways to choose 2 out of 12 tosses to get each number, and then multiply it by the number of permutations of the remaining 10 tosses.
For each number (1, 2, 3, 4, 5, 6):
Number of ways = (12 choose 2) * (10!/2!)
Step 3: Calculate the probability for each number.
The probability of each number occurring exactly twice can be calculated by dividing the number of ways to achieve it by the total possible outcomes.
For each number (1, 2, 3, 4, 5, 6):
Probability = Number of ways / Total possible outcomes
It can be simplified as follows:
Probability = (12 choose 2) * (10!/2!) / 2,176,782,336
Calculating each probability individually will give you the probability of getting a 1, 2, 3, 4, 5, or 6 exactly twice when a fair die is tossed 12 times.