Once I heard another example: how many faces will have the union of a pyramid based on a square whose faces are equilateral triangles and a regular tetrahedron that is glued to one of these faces: it was said that the right answer to this problem had a negative correlation with the result of the entire test… ]]>

1) Let S be a finite multiset of integers (i.e., I can have repetitions) with mean 6. Tina randomly selects a number from S, and Sergio randomly selects a number between 1 and 10. Let p be the probability that Sergio picks a larger number than Tina. As S varies, what is the range of possible values of p?

2) Let S be a finite set of integers with mean 3. Tina randomly selects two distinct numbers from S, and Sergio randomly selects a number between 1 and 10. Let p be the probability that the number Sergio picks is larger than the sum of the numbers Tina picks. As S varies, what is the range of possible values of p?

]]>1/2 wins or ties 3/10,

but rather

1+2 wins or ties 3/10 of the time. ]]>

1/2 wins or ties 3/10

1/3 wins or ties 4/10

1/4 wins or ties 5/10

2/3 wins or ties 5/10

1/5 wins or ties 6/10

2/4 wins or ties 6/10

2/5 wins or ties 7/10

3/4 wins or ties 7/10

3/5 wins or ties 8/10

4/5 wins or ties 9/10

As each pair occurs with probability 1/10, we get (3+4+5+5+6+6+7+7+8+9)/10/10 = 60/100

Same problem, slightly altered, average still 6, but answer is not 6/10

If we change the first set to {0,1,3,4,7} the average holds, but the result falls.

0/1 wins or ties 1/10

0/3 wins or ties 3/10

0/4 wins or ties 4/10

1/3 wins or ties 4/10

1/4 wins or ties 5/10

0/7 wins or ties 7/10

3/4 wins or ties 7/10

1/7 wins or ties 8/10

3/7 wins or ties 10/10

4/7 wins or ties 10/10

As each pair occurs with probability 1/10, we get (1+3+4+4+5+7+7+8+10+10)/10/10 = 59/100

Discussion:

In the first example, the probability that a sum is greater than or equal to a random number from 1 to 10 is directly proportional to the sum (all sums are between 3 and 10). In the second example, one sum is greater than 10, breaking the symmetry that previously kept the probability proportional to the sum.