We consider a model of binary opinion dynamics where one opinion is inherently 'superior' than the other and social agents exhibit a 'bias' towards the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability $α>0$ irrespective of its current opinion and the opinions of the other agents. With probability $1-α$ it adopts the majority opinion among two randomly sampled neighbours and itself. We are interested in the time it takes for the network to converge to a consensus state where all the agents adopt the superior alternative. In a fully connected network of size $n$, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $Θ(n \log n)$ when the bias parameter $α$ is sufficiently high, i.e., $α> α_c$ where $α_c$ is a threshold parameter that is uniquely characterised. When the bias is low, i.e., when $α\in (0,α_c]$, we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(α)$. If this is not the case, then we show that the network takes $Ω(\exp(Θ(n)))$ time on average to reach consensus. Through numerical simulations we observe similar behaviour for other classes of graphs.