What is the probability that the die shows an odd number or a number less than 3 on top?

So we're first assuming that it's a standard die, one with only six sides.

Now, we can first look at the statement, odd number.

There are three possibilities for this.

1, 3, and 5.

Now for, less than 3.

Two possible numbers.

1, and 2.

So the possible numbers that would satisfy these conditions would be, 

1, 2, 3, and 5.

So 4 numbers.

Since there are 6 total possibilities, the fraction would be 4/6, or 2/3.

So the probability is 2/3.