Some things in Mathematics are counterintuitive, well at least surprising. The first thing I think of is that there are different 'infinities'. Try to explain to your neighbor that the number of even numbers is equal to the number of numbers divisible by three. Or that there are more points on the ( real number ) line between 0 and than there are integers.
Or this one. Random times is Random not Random? Not uniformly random to be precise. In the experiment below I threw a coin with sides 0 and 1 99,999 times. The first graph shows the number of 0s and 1s thrown. Practically equal at 50,000. Then I did the same with two coins 99,999 times but multiplied the product ( mod 2 ). The second graph shows the result. 75,000 zero's and 25,000 1s.
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Now imagine the same for any number. A number ends with 0,1,2,..., or 9. The number of 0s,1s, 2s will be equal if you take a large set of random numbers. But if you multiply two random numbers the last digits will not have a uniform distribution.
Interesting is that the distribution depends on the type of operation.
Two comments.
ReplyDelete1) Your Mathematica code can be made faster:
Tally@(RandomInteger[1, {10^7}])
Tally@(RandomInteger[1, {10^7}] RandomInteger[1, {10^7}])
2) The results make sense. If you look at the Tally results, for multiplying them together, 25% are 1, 75% are zero. This is because 0*0=0, 0*1=0, and 1*0=0, while only 1*1=1. So, on average, 3/4 of the results will be zero and 1/4 will be 1.
1) Did not know Tally, thanks, interesting.
ReplyDelete2) My conclusion too. Therefore the graphs.