Question

Suppose the total of rolling any particular number of such dice is normally distributed, with a standard deviation and mean as shown

If you are rolling 4 dice, the standard deviation is approximately 3.416

If you are rolling 5 dice, the standard deviation is approximately 3.819

If you are rolling 6 dice, the standard deviation is approximately 4.183

Also,

If you are rolling 4 dice, the mean is approximately 14

If you are rolling 5 dice, the mean is approximately 17.55

If you are rolling 6 dice, the mean is approximately 21

If you rolled 4 dice, what is the theoretical probability you would find that in 1000 rolls you obtained a total exactly 2 less than the mean? That is, for rolling FOUR dice your total would be 12 or less; for FIVE dice your total would be 15.55 or less; for SIX dice your total would be 19 or less.

Show the computations needed to answer this question! You need to calculate the z-score for the indicated total and then reference the standard normal distribution table.

Answer 1

Rolling 4 dice

Suppose, random variable X denotes total obtained in rolling 4 dice.

Required probability is given by

Rolling 5 dice

Suppose, random variable Y denotes total obtained in rolling 5 dice.

Required probability is given by

Rolling 6 dice

Suppose, random variable W denotes total obtained in rolling 6 dice.

Required probability is given by