A computerstore has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance of the number of ink jet printers.
To find the mean and variance of the number of ink jet printers in a group of 10 chosen at random from the store, we need to use probability theory and some basic calculations.
Let's start by calculating the mean (or expected value) of the number of ink jet printers.
Mean (Expected value):
The mean is calculated by multiplying each possible outcome by its corresponding probability and then summing them up. In this case, we want to find the mean number of ink jet printers in a group of 10.
The probability of selecting an ink jet printer is given by the ratio of ink jet printers to the total number of printers in the store:
P(ink jet printer) = 50 / 75 = 2/3
Now, let's calculate the mean:
Mean = (number of trials) * (probability of success)
Mean = 10 * (2/3) = 20/3 ≈ 6.67
So, the mean number of ink jet printers in a group of 10 chosen at random is approximately 6.67.
Now let's move on to calculating the variance.
Variance:
The variance measures the spread or dispersion of a random variable. It is calculated by summing the squared difference between each outcome and the mean, multiplied by their probabilities.
We know that the mean number of ink jet printers is 6.67, so we need to calculate the variance using this mean and the probability of selecting an ink jet printer (2/3).
Variance = (number of trials) * (probability of success) * (probability of failure)
Variance = 10 * (2/3) * (1 - 2/3)
Variance = 10 * (2/3) * (1/3)
Variance = 20/9 ≈ 2.22
Therefore, the variance of the number of ink jet printers in a group of 10 chosen at random is approximately 2.22.