We study the length of T -contaminated runs of heads in the well-known coin tossing experiment. A T -contaminated run of heads is a sequence of consecutive heads interrupted by T tails. For T = 1 and T = 2 we find the asymptotic distribution for the first hitting time of the T contaminated run of heads having length m; furthermore, we obtain a limit theorem for the length of the longest T -contaminated head run. We prove that the rate of the approximation of our accompanying distribution for the length of the longest T -contaminated head run is considerably better than the previous ones. For the proof we use a powerful lemma by Csáki, Földes and Komlós, see [1].