Let $\{X_n\}^{\infty}_0$ be a Markov chain with values in $[0,1]$
generated by the iteration of random logistic maps defined by
$X_{n+1}=f_{C_{n+1}}(X_n)\equiv C_{n+1}X_n(1-X_n)$, $n=0,1,2,\ldots\,$, with
$\{C_n\}^{\infty}_1$ being independent and identically distributed random
variables with values in $[0,4]$ and independent of $X_0$. This paper provides
a class of examples where $C_i$ take only two values $\lambda$ and $\mu$ such
that there exist two distinct invariant probability distributions $\pi_0$ and
$\pi_1$ supported by the open interval $(0,1)$. This settles a question raised
by R. N. Bhattacharya.