Lava Lamp Arrangement Odds
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?
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
There are $ \binom{6}{3}=20 $ ways for Ryan to arrange the lamps, and $ \binom{6}{3}=20 $ ways for him to choose which lamps are on, giving $ 20\cdot20=400 $ total possible outcomes. There are two cases for the desired outcomes: either the left lamp is on, or it isn't. If the left lamp is on, there are $ \binom{5}{2}=10 $ ways to choose which other lamps are on, and $ \binom{5}{2}=10 $ ways to choose which other lamps are red. This gives $ 10\cdot10=100 $ possibilities. If the first lamp isn't on, there are $ \binom{5}{3}=10 $ ways to choose which lamps are on, and since both the leftmost lamp and the leftmost lit lamp must be red, there are $ \binom{4}{1}=4 $ ways to choose which other lamp is red. This case gives 40 valid possibilities, for a total of 140 valid arrangements out of 400. Therefore, the probability is $ \dfrac{140}{400}=\boxed{\dfrac{7}{20}} $.
Freeflows