Question

. Three numbers are chosen at random from the set {1, 2, . . . , 50}. Find the probability
that the chosen numbers are in geometric progression.

Answer 1

What GP's are possible? If the 3 numbers are chosen independently, then we have

1,1,1
1,2,4
1,3,9
1,4,16
1,5,25
1,6,36
1,7,49

2,2,2
2,4,8
2,6,18
2,8,32
2,10,50

3,3,3
3,6,12
3,9,27
3,12,48

4,4,4
4,8,16
4,12,36

5,5,5
5,10,20
5,15,45

6,6,6
6,12,24

and so on.
If the three numbers are chosen without replacement, like picking three cards from a deck, then the number of possible GP's is much smaller.

Anyway, once you have the set of GP's you can figure the probability of each, then just add them up.

Answer 2

0.00122