ANSWER
1/35
EXPLANATION
First we have to find how many ways you can pick 3 horses out of 7. For the first place you have 7 horses that can be there. For the second place, you have 6 horses (you already placed one horse) and for the third place you have 5 horses. The number of ways to pick 3 horses - in order - from 7 horses is:
[tex]7\times6\times5=210[/tex]Now we want to know the permutations for 3 horses. Once you have picked one combination out of the 210 combinations of 3 horses, there are many ways to arrange them. This is:
[tex]3!=3\times2\times1=6[/tex]Let 'A' be the event where the gambler wins his bet:
[tex]P(A)=\frac{6}{210}[/tex]We can simplify this fraction:
[tex]P(A)=\frac{1}{35}[/tex]