Solved – The mean value of a selected die from an infinite series of rolls

If I roll a pair of dice an infinite number of times, and always select the higher value of the two, will the expected mean of the highest values exceed 3.5?

It would seem that it must be since if I rolled a million dice, and selected the highest value each time, the odds are overwhelming that sixes would be available in each roll. Thus, the expected mean would have to be something like 5.999999999999…

However, I can't seem to figure out what the expected value would be with my example using just 2 dice. Can someone help me arrive at a number? Would it barely exceed 3.5? Is this even something that can be calculated?

The experiment can also be simulated. This approach is useful when enumeration is difficult (like rolling 3 dice).

# fix the seed for reproducibility set.seed(123)  # simulate pair of dice rolls = matrix(sample(1:6, 2000000, replace=T), ncol=2)  # compute expected value mean(apply(rolls, 1, max)) [1] 4.471531 

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