Proof for the Question (i)
There are five rows and six numbers.
By the Pigeonhole Principle, at least two numbers belong to the same row.
Hence at least two numbers have the same first digit.
Proof for the Question (ii)
How about the last digits?
Observe that in about 90% of the draws, at least 2 drawn numbers having the same last digit. Is the machine biased ?
Let us find p, the probability of having at least two last digits equal.
Suffice to know q, the probability of having all the last digits distinct as p = 1 - q.
Assume we have 50 balls and the probability of getting each ball is the same.
q = 50/50 x 45/49 x 40/48 x 35/47 x 30/46 x 25/45 x 20/44 = 0.0938
Therefore, p = 1- 0.0938= 0.906 which matches with the observation (90%).
Conclusion
Next time when you buy Mark Six, may be it is better to choose
numbers such that at least two of them have the same last digit.
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