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!!!!

Guest Jan 2, 2021

#1**0 **

Happy new year!

Perhaps we can simplify the cases. We know that the probability the leftmost lamp is red is 1/2, so we just multiply our final answer by that.

Case 1: leftmost lamp is on (and obviously red)

5 choose 2/ 6 choose 3 = 1/2

Case 2: the second lamp is the first on and is red

1/2 * 3/5 *2/5 = 3/25

Case 3: the third lamp is the first on and is red

1/2 *2/5 *3/4 * 2/5 = 3/50

Case 4: the fourth lamp is the first on and is red

1/2 * 2/5 *1/4* 1 *2/5 = 1/50

Not sure where you went wrong... though.

Pangolin14 Jan 2, 2021