<em>Y</em>₁ and <em>Y</em>₂ are independent, so their joint density is
By definition of conditional probability,
P(<em>Y</em>₁ > <em>Y</em>₂ | <em>Y</em>₁ < 2 <em>Y</em>₂) = P((<em>Y</em>₁ > <em>Y</em>₂) and (<em>Y</em>₁ < 2 <em>Y</em>₂)) / P(<em>Y</em>₁ < 2 <em>Y</em>₂)
Use the joint density to compute the component probabilities:
• numerator:
• denominator:
(I leave the details of the second integral to you)
Then you should end up with
P(<em>Y</em>₁ > <em>Y</em>₂ | <em>Y</em>₁ < 2 <em>Y</em>₂) = (1/6) / (2/3) = 1/4