Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on. What is the probability that the leftmost lamp on the shelf is red?

The probability that the leftmost lamp on the shelf is red is, as Melody said above, \(\frac 1 2\)

Let's take a (somewhat) closer look, however,

We have 6 lamps, 3 are red, 3 are blue. The shelf looks like this:

______

Now, we have 2 ways to fill the leftmost shelf, either with a red or a blue. Here is a basic formula in probability:

\(\frac{\text{# of desired outcomes}}{\text{# of all possible outcomes}}\)

Since we have 3 desired outcomes (a red) and 6 possible outcomes we get:

\(\frac{3}{6} = \frac{1}{2}\)

This is half, as Melody said.