$A$ die has its six faces marked ${0, 1, 1, 1, 6, 6}$. Two such dice are thrown together and the total score is recorded.
$(i)$ How many different scores are possible?
$(ii)$ What is the probability of getting a total of $7$?

(C) Given,a die has its six faces marked ${0, 1, 1, 1, 6, 6}$.
Total sample space,$n(S) = 6^2 = 36$.
$(i)$ The possible sums are obtained by adding the values on the two dice:
$0+0=0, 0+1=1, 0+6=6, 1+1=2, 1+6=7, 6+6=12$.
Thus,the different scores possible are ${0, 1, 2, 6, 7, 12}$. There are $6$ possible scores.
$(ii)$ Let $E$ be the event of getting a sum of $7$.
The pairs $(d_1, d_2)$ that result in a sum of $7$ are:
$(1, 6)$ occurs $3 \times 2 = 6$ times.
$(6, 1)$ occurs $2 \times 3 = 6$ times.
Total favorable outcomes $n(E) = 6 + 6 = 12$.
$P(E) = \frac{n(E)}{n(S)} = \frac{12}{36} = \frac{1}{3}$.