= Solution
{c}
Let $R$ denote the red die. Conditional on $R=r$, each white roll succeeds independently with <probability> $p_r=(7-r)/6$. Consequently $T\mid R=r$ has the <geometric distribution> on $\{1,2,\ldots\}$ and <expected value> $1/p_r=6/(7-r)$. The <law of total expectation> gives
$$
\boxed{\mathbb E T=\frac16\sum_{r=1}^6\frac6{7-r}
=\sum_{k=1}^6\frac1k=\frac{49}{20}.}
$$
This is $2.45$ rolls; the averaging must be over conditional <expected values>, not over success <probabilities> before taking a reciprocal.
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