Why do people recommend wider tyres in car for better road grip? Tyre companies boast of their wider tires for better grip on road. Also, the F1 cars have broad tires for better grip. But as far as I know Friction does not depend on the surface area of contact between the materials. Even the formula says so.. 
$F=\mu mg$ (where $F$ = Force of friction, $\mu$ = coefficient of friction, $m$ = mass, $g$ = gravity)
Can anyone please tell me the relation between broad tires and road grip ?
 A: It's a surprisingly complicated question. Given your mention of friction, probably the main point is that for a car tyre the friction is not linearly dependant on load. Wikipedia has some information about this here.
If you had perfectly smooth surfaces the friction is actually proportional to the area of contact and independant of the load. This is because friction is an adhesive effect between atoms/molecules on the surfaces that are in contact. However in the real world surfaces are not smooth. If you touch two metal surfaces together the contact is between high spots on the two surfaces so the area that is in contact is much less than than the apparent area of contact. If you increase the load you deform these high spots and broaden them, so the effect of load is to increase the real area of contact. The real area of contact is approximately proportional to the load, and the friction is proportional to the area of contact, so the friction ends up being approximately proportional to the load.
However a rubber type is a lot softer than metal, and a road is a lot rougher than a metal plate. Even at low loads the tyre deforms to key into the irregularities in the road, so increasing the load has a lesser effect. That's why you get the sub-linear dependance described in the Wikipedia article.
But this is only the start of the complexity. If you use a wider tyre the contact patch area isn't necessarily bigger. A wider tyre has a wider shorter contact patch while a narrow tyre has a narrower longer contact patch. The contact patch area depends on the tyre pressure, the deformation of the sidewalls and probably lots of other things I can't think of at the moment.
And anyway, if by "grip" you mean grip when cornering, the grip isn't just controlled by the contact patch area. When a car is cornering the contact patch is being twisted. This is known as the slip angle. The wider shorter contact patch on a wide tyre has a smaller slip angle and as a result grips better.
A: The answer by @JohnRennie unfortunately misses the main reason why wider tires are recommended for better friction:

It is true that wider tires commonly have better traction. The main
reason why this is so does not relate to contact patch, however, but
to composition. Soft compound tires are required to be wider in order
for the side-wall to support the weight of the car. Softer tires have
a larger coefficient of friction, therefore better traction. A narrow,
soft tire would not be strong enough, nor would it last very long.
Wear in a tire is related to contact patch. Harder compound tires wear
much longer, and can be narrower. They do, however have a lower
coefficient of friction, therefore less traction. Among tires of the
same type and composition, here is no appreciable difference in
'traction' with different widths. Wider tires, assuming all other
factors are equal, commonly have stiffer side-walls and experience
less roll. This gives better cornering performance.
Answered by: Daryl Garner, M.S., Physics teacher MacArthur High
School, Lawton, OK

A: The reason why wider tires are chosen over thiner ones is not because of greater friction. And the reason is not complicated, or does not "start a complexity".. the reason why wider tires are chosen is because they last longer than thin ones! Nothing about friction at all. Racers would always have to replace thiner ones, so they figured, why not get larger ones! 
