Two different dice are thrown simultaneously. Find the probability of getting :

(i) a number greater than 3 on each dice

(ii) an odd number on both dice.

When two different dice are rolled together, the total number of outcomes is 6 × 6 i.e. 36 and all outcomes are equally likely. The sample space of the random experiment has 36 equally likely outcomes. The sample of the experiment

S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
       (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
       (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
       (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
       (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
       (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6).}

It consists of 36 equally likely outcomes.

(i) Let A be the event of getting 'a number greater than 3 on each dice', then

A = {(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}.

∴ The number of outcomes favourable to event A = 9.

∴ P(a number greater than 3 on each dice) = $\dfrac{9}{36} = \dfrac{1}{4}.$

Hence, the probability of getting a number greater than 3 on each dice is $\dfrac{1}{4}$.

(ii) Let B be the event of getting 'an odd number on both dice', then

A = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}.

∴ The number of outcomes favourable to event B = 9.

∴ P(an odd number on both the dice) = $\dfrac{9}{36} = \dfrac{1}{4}.$

Hence, the probability of getting an odd number on both the dice is $\dfrac{1}{4}$.