You know that H is dominant for height locus and H is recessive. Is the F2 progeny are a mixture of 3:1 dominant: recessive and you pick a dominant plant at random what will be the probability that the F3 will have a mixture of dominant and recessive traits as well?

Here we are actually selecting the plants which are heterozygous for the trait as in Hh. Only selfing these will give both dominant and recessive trait in next generation. As we are not taking hh into consideration the ratio resulting is 2/3.