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Bell-shaped distribution Two dice roll: We now roll two dice of white and blue, and add up the pips of faced-up sides. The smallest sum ‘2’ is given by ( Table 3. Sum of the facing up pips of two dice roll
Since there are in all 36 events in table 3, the probabilities of sum ‘2’ and ‘12’ are 1/36, sum ‘3’ and ‘11’ are 2/36, sum ‘4’ and ‘10’ are 3/36, and so on, as they appear in the bar graph of figure 3 over the sample space {2, 3, 4, ..., 11, 12}. Here comes a surprise. The uniform distribution (figure 2) of one die roll has become a triangle distribution (figure 3) when two dice are rolled and the faced-up pips are added up. That is, you are more likely to get sum ‘7’ than any other sums in two dice throw. With Prog#7c you can check how closely a triangle distribution is realized by increasing the number of dice throws up to 5,000.
Figure 3. Two dice throw
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, 
) and
the largest sum is 12 when ( 
, 
), so
that {2, 3, 4, …, 11, 12} is the sample space. But, sum
‘3’ can result from two events ( 
) and (
,
, 
) will
give sum ’11.’ In table 3, we list all possible events
for the sums of {2, 3, 4, …, 11, 12} of a two dice roll. 
), (
, 
), (
, 