You are given a machine and told that the machine is a random number generator for the numbers between 1 and 10. You generate 1,000 numbers with the machine. Which result below would most strongly lead you to the conclusion that the machine is not "fair"?

30% of the numbers are less than 3

40% of the numbers are greater than 4 but less than 9

50% of the numbers are greater than 5

60% of the numbers are less than or equal to 6

Question

Rodrigo

Mathematics

30 April, 19:00

You are given a machine and told that the machine is a random number generator for the numbers between 1 and 10. You generate 1,000 numbers with the machine. Which result below would most strongly lead you to the conclusion that the machine is not "fair"?

30% of the numbers are less than 3

40% of the numbers are greater than 4 but less than 9

50% of the numbers are greater than 5

60% of the numbers are less than or equal to 6

+5

Answers (1)

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Donaldson · Answer 1

Our sample space is composed by 10 numbers, 1 to 10. Let's see how many numbers satisfy each statement, to understand how "big" each subset is with respect to the whole space:

There are two numbers which are less than 3: 1 and 2. This means that the subset identified by "numbers that are less than 3" is composed by 2 numbers. Since there are 10 numbers in total, this subspace represents the 2/10 = 0.2 = 20% of the whole space, and thus we should expect 20% of the numbers to be less than 3, not the 30%.

There are four numbers which are greater than 4 but less than 9: 5, 6, 7 and 8. This means that the subset identified by "numbers that are greater than 4 but less than 9" is composed by 4 numbers. Since there are 10 numbers in total, this subspace represents the 4/10 = 0.4 = 40% of the whole space, and thus we should expect 40% of the numbers to be less greater than 4 but less than 9. We indeed have this value, so the machine seems fair.

There are five numbers which are greater than 5: 6, 7, 8, 9 and 10. This means that the subset identified by "numbers that are greater than 5" is composed by 5 numbers. Since there are 10 numbers in total, this subspace represents the 5/10 = 0.5 = 50% of the whole space, and thus we should expect 50% of the numbers to be greater than 5. We indeed have this value, so the machine seems fair.

Finally, there are six numbers which are less than or equal to 6: 1, 2, 3, 4, 5 and 6. This means that the subset identified by "numbers that are less than or equal to 6" is composed by 6 numbers. Since there are 10 numbers in total, this subspace represents the 6/10 = 0.6 = 60% of the whole space, and thus we should expect 60% of the numbers to be less than or equal to 6. We indeed have this value, so the machine seems fair.

So, the only observed probability that doesn't match with the result is the first one.