18.The entire sample space is seen below. Note again that there are 36 different (unique) outcomes. Importantly, if all of the probabilities were added from this entire sample space, the total would add up to one. Basic probability for rolling dice The table below shows all of the possible scenarios listed in the sample space (36 total). Sum of two dice # of possible ways to get sum 2 3 4 5 6 7 8 1 2 3 4 5 6 5

9 10 11 12 Notice that there are 6 ways to obtain the sum of 7 from a pair of dice. Thus we expect the probability to achieve a sum of 7 to be the greatest. What would be the probability of rolling a sum of 2? 4 3 2 1 Let us assume there is a large sample of data collected by (many) rolls. The mean value is simply the center of the distribution of values within the sample. Let there be Ntrials of a measurement of a quantity x. The mean value of the measured quantity xover Ntrials can be written as .11NiixNx(1) This is an equation that simply states: For the “best” value of a measurement of some quantityx, it is simply the arithmetic average xof an Nnumber of trials. Suppose you were rolling a single piece of die. There are 6 possible outcomes. When rolling two or more dice many times, the mean value from all N-rolls, and all possible outcomes (?) that have a frequency of occurrence can be written as .)(11NiiixxfNx(2) Two other important quantities are variance and standard deviation. Variance is defined as the arithmetic average of the square of the difference between each trial and the mean. That is, it is a measure as to how far a random number is from the mean. The standard deviation (𝜎𝑥) or the square root of the variance (𝜎2)is expressed as:

NiiNixxxNdN1212)(11)(11. (3) For the above equation, the quantity xi is the value for the ithtrial. The deviation or the residual is defined as the difference between the ithtrial andx, di= xi- Let us assume that there is some fluctuation within an Nnumber of trials. If Nsufficiently large, then the fluctuations (the spread) would follow a Gaussian (Normal) distribution. The figure below shows what this would look like with a large N-number of trials. The width of this curve is simply the standard deviation. Notice the percentages of a large data set that fall within the inflection points. For example, 68.2 % of the data fall within one standard deviation from the mean value. The equation describing the normal distribution is written as ?(𝑥) =1𝜎√2𝜋?[−(𝑥−𝑥̅)22𝜎2].(4) For a large collection of data points, the values typically cluster around the mean value, if the precision of the test is well within acceptable means. In a word, since the standard deviation is a measure of the width of the above distribution curve, a large standard deviation correlates to measurements of low precision. Rolling dice is a measurement, which is random in nature. If there are two dice, one expects the average to float around the sum of seven. The values will of course fluctuate above

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