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Two dice are thrown at the same time. Determine the probability that the (i) sum of the numbers on the two dice is $5$, and (ii) difference of the numbers on the two dice is $3$.
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Sol. Total outcomes $= 36$ (1/2 Mark)
(i) Number of outcomes with sum of the numbers on the two dice is $5 = 4$
$ (1,4) (4,1) (2,3) (3,2)$ (1 Mark)
P(sum of the numbers on two dice is $5$) $= \frac{4}{36}$ or $\frac{1}{9}$ (1/2 Mark)
(ii) Number of outcomes with difference of the numbers on the two dice is $3 = 6$
$ (1,4) (4,1) (5,2) (2,5) (6,3) (3,6)$ (1 Mark)
P(difference of the numbers on the two dice is $3$) $= \frac{6}{36}$ or $\frac{1}{6}$ (1/2 Mark)
(i) Number of outcomes with sum of the numbers on the two dice is $5 = 4$
$ (1,4) (4,1) (2,3) (3,2)$ (1 Mark)
P(sum of the numbers on two dice is $5$) $= \frac{4}{36}$ or $\frac{1}{9}$ (1/2 Mark)
(ii) Number of outcomes with difference of the numbers on the two dice is $3 = 6$
$ (1,4) (4,1) (5,2) (2,5) (6,3) (3,6)$ (1 Mark)
P(difference of the numbers on the two dice is $3$) $= \frac{6}{36}$ or $\frac{1}{6}$ (1/2 Mark)