Let $\nu_\lambda^p$ be the distribution of the random series
$\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random
variables taking the values 0,1 with probabilities $p,1-p$. These measures are
the well-known (biased) Bernoulli convolutions.

In this paper we study the multifractal spectrum of $\nu_\lambda^p$ for
typical $\lambda$. Namely, we investigate the size of the sets

\[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log
\nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \]

In this paper we compute the multifractal analysis for local dimensions of
Bernoulli measures supported on the self-affine carpets introduced by
Bedford-McMullen. This extends the work of King where the multifractal analysis
is computed with strong additional separation assumptions.