let's say you picked the door with the "goat". you have a 1/4 chance of doing this. in this case, switching gives you a 1/2 chance of winning the prize.

let's say you picked the door with the prize. you also have a 1/4 chance of doing this. in this case, swtiching gives you a 0 chance of winning the prize.

finally, let's say you picked a "empty" door. you have a 1/2 chance of doing this. in this case, switching gives you a 1/2 chance of winning the prize.

so you total chance of winning the prize is: (1/4)(1/2) + (1/4)(0) + (1/2)(1/2) = 3/8.

now let's analyze the "don't switch" strategy:

you pick the door with the "goat (booby prize)". you have a 1/4 chance of doing this, and in this case, you never win.

you pick the door with the prize. you have a 1/4 chance of doing this, and in this case, you always win.

you pick an empty door. you have a 1/2 chance of doing this, and in this case, you never win.

total chance of winning the prize is: (1/4)(0) + (1/4)(1) + (1/2)(0) = 1/4.

it's still better to switch.