Calculate the mean and standard deviation of the probability distribution created by rolling a dice. Either show work or explain how your answer was calculated.
To calculate the mean and standard deviation of the probability distribution created by rolling a dice, we first need to determine the probabilities of each possible outcome.
When rolling a standard six-sided dice, each face has equal probability of 1/6. Therefore, the probability of getting any particular number on the dice is 1/6.
Let's calculate the mean:
The mean (μ) of a probability distribution is calculated by multiplying each possible outcome by its probability and summing them up.
The possible outcomes when rolling a dice are 1, 2, 3, 4, 5, and 6. So the calculation for the mean is:
μ = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)
= (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6)
= 21/6
= 3.5
Therefore, the mean of the probability distribution created by rolling a dice is 3.5.
Now let's calculate the standard deviation:
The standard deviation (σ) of a probability distribution is a measure of the spread of the distribution. It is calculated by taking the square root of the sum of the squared differences between each possible outcome and the mean, multiplied by their probabilities.
The calculation for the standard deviation is as follows:
σ = √(((1 - 3.5)^2 * 1/6) + ((2 - 3.5)^2 * 1/6) + ((3 - 3.5)^2 * 1/6) + ((4 - 3.5)^2 * 1/6) + ((5 - 3.5)^2 * 1/6) + ((6 - 3.5)^2 * 1/6))
= √((2.5^2 * 1/6) + (1.5^2 * 1/6) + (0.5^2 * 1/6) + (0.5^2 * 1/6) + (1.5^2 * 1/6) + (2.5^2 * 1/6))
= √(6.25/6 + 2.25/6 + 0.25/6 + 0.25/6 + 2.25/6 + 6.25/6)
= √(17.5/6)
≈ √2.92
≈ 1.71
Therefore, the standard deviation of the probability distribution created by rolling a dice is approximately 1.71.