Why is induced drag less on a high span wing?
You start from wrong assumptions, which explains your doubts. The line 


  the induced drag is due to the tip vortices


is as true as saying that wet streets cause rain. Also, the opinion that the


  tip vortices strength will be as same as the bound vortex


is wrong. Unfortunately, many authors don't understand the topic themselves and copy what others have written before without thinking the issue through. Ideally, you would forget about all what you have heard about vortices and lifting lines, but since you ask I will try to explain potential flow theory a little.

In potential flow theory, lift is caused by vortices which are caused by the movement of a wing through air. These vortices run along a closed line: Within the wing they form the bound vortex, then they leave the wing backwards as trailing vortices and are connected at the point where the movement started by the starting vortex.

Now comes the important part which most authors conveniently leave out: There is no single vortex; instead, potential flow assumes an infinite number of infinitesimally small vortices which form out of nowhere when lift is increased or speed is reduced. Consequently, no single vortex leaves the wing at the tips, instead, a sheet of vortices leaves the wing at the trailing edge. The change in strength of the bound vortices over span is equivalent to the strength of the vortices leaving the wing, so the vortices fade out towards the tips.

My advice is: If you do not want to operate or to write a potential flow code, do yourself a favor and forget all that. It is much better to interpret lift as the consequence of a pressure field around a wing which accelerates the air flowing around this wing downwards. Induced drag is simply the component of the resulting pressure forces parallel to the direction of movement, while the perpendicular component is lift. Please make sure to follow at least the last link; it gives a very good explanation what induced drag really is.

Tip vortices are the consequence of air filling the void above the downward moving air behind the wing. They are not originating from the wingtips, but the consequence of the vortex sheet rolling up (if you want to stay in that picture). Note that the distance between the cores of the vortices is much smaller than the wingspan. For an elliptic wing of span $b$, it is actually only $\frac{\pi}{4}\cdot b$

A higher wingspan allows to capture more air for lift creation, so less downward acceleration is needed. Lower downwash speed also causes a less powerful trailing vortex. Note that the mass of air affected by the wing grows with the square of the wingspan!