Calculating probability of getting at least 3 red ball drawn from a bag having 6 red ball,5 blue,4 green in 5th attempt.

If you try to sum up the probabilities of drawing exactly $3$ red balls on the third, fourth or fifth attempt, you will find it a bit tricky because you'll always have to keep in mind that the last trial $(say, k),$must be a win, and work out the combinations for $k-1$ trials followed by a win on the $k_{th}$ trial, which can become messy.

But if you understand that in $5$ attempts, you can have at most $2$ failures, then it becomes very simple.

$dbinom50 q^0p^5 + dbinom51 q^1p^4 +dbinom52 q^2p^3$, and evaluate