Help on Alcumus Problem

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, and the leftmost lamp which is turned on is also red?

 

I have found that the answer is 7/20. However my answer is 27/80. 

 

Here's what I'm thinking :

 

We should do some casework here.

 

Case 1 : the leftmost lamp is red and it is on -- prob : (3/6 * 1/2) = 1/4

 

Case 2 : the leftmost lamp is red and off & the next one is red and on -- prob : (3/6 * 1/2) * (2/5 * 1/2) = 1/20

 

Case 3 : the leftmost lamp is red and off & the next one is red and off & the one after that is red and on -- prob : (3/6 * 1/2) * (2/5 * 1/2) * (1/4 * 1/2) = 1/160

 

Case 4 : the leftmost lamp is red and off & the next one is blue and off & the one after that is red and on -- prob : (3/6 * 1/2) * (3/5 * 1/2) * (1/2 * 1/2) = 3/160

 

Case 5 : the leftmost lamp is red and off & the next one is blue and off & the one after that is blue and off & the one after that is red and on -- prob : (3/6 * 1/2) * (3/5 * 1/2) * (1/2 * 1/2) * (2/3 * 1/2) = 1/160

 

Case 6 : the leftmost lamp is red and off & the next one is red and off & the one after that is blue and off & the one after that is red and on -- prob : (3/6 * 1/2) * (2/5 * 1/2) * (3/4 * 1/2) * (1/3 * 1/2) = 1/320 

 

Case 7 : the leftmost lamp is red and off & the next one is blue and off & the one after that is red and off & the one after that is red and on -- prob : (3/6 * 1/2) * (3/5 * 1/2) * (1/2 * 1/2) * (1/3 * 1/2) = 1/320

 

 

Now we add all of these up...

 

1/4 + 1/20 + 1/160 + 3/160 + 1/160 + 1/320 + 1/320 = 27/80

 

I can't seem to figure out what I did wrong here...did I skip any cases???

 

My answer is only 1/80 away from the correct one..

 

Any help is greatly appreciated!

 

Thank You, and Happy New Year 2021!!!!