Calculating Mean and Standard Deviation for a Biased Die Rolled 2000 Times

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The problem involves a biased die with a probability of rolling a six at 1/4, rolled 2000 times. The mean number of sixes, denoted as X, is calculated as 500, derived from the formula for the mean of a binomial distribution (2000 * 1/4). For the standard deviation of X, it is necessary to use the formula for the standard deviation of a binomial distribution, which is the square root of the product of the number of trials and the probabilities of success and failure. The standard deviation can be expressed as a surd, specifically √(2000 * (1/4) * (3/4)). Understanding these calculations is essential for accurately determining the statistical properties of the biased die rolls.
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Homework Statement



A die is biased such that the probability of getting a six is 1/4. The die is rolled 2000 times. Let X be the number of sixes obtained. Find,

a) the mean of X
b) the standard deviation of X, leaving your answer as a surd.

Homework Equations





The Attempt at a Solution



a) The mean of X is simple, given P(6)=1/4, and the dice is rolled 2000 times, the mean of X is 2000(1/4)=500

b) This is where I don't know what to do. Any guidance?
 
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