Yes, the uniform probability distribution is symmetric about the
mode. Draw the sketch of the uniform probability distribution. If
we say that the distribution is uniform, then we obtain the same
constant for the continuous variable.
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The uniform probability distribution is one in which the
probability is the same throughout its domain, as stated above. By
definition, then, there can be no value (or sub-domain) for which
the probability is greater than elsewhere. In other words, a
uniform probability distribution has no mode. The mode does not
exist.
The distribution cannot, therefore, be symmetric about something
that does not exist.