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jmc

counting and probability senior

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?
Solution
There are ways for Ryan to arrange the lamps, and ways for him to choose which lamps are on, giving 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 ways to choose which other lamps are on, and ways to choose which other lamps are red. This gives possibilities. If the first lamp isn't on, there are ways to choose which lamps are on, and since both the leftmost lamp and the leftmost lit lamp must be red, there are 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 .
Final answer
\dfrac{7}{20}