**Q) In a two-dice game, a player throws two dice simultaneously. A player scores the sum of the two dice thrown and gradually reaches a higher score as they continue to roll. Answer the following questions:**

**i. Find the probability that the difference between the numbers on the two dice is 3.**

**ii. Find the probability that the product of the numbers on the two dice is more than 18.**

**Ans: **

Total possible outcomes with two dice are thrown together:

(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),

∴ Total outcomes = 36

**(i) For the difference of 3, the possible outcomes are:**

(1, 4), (2, 5), (3, 6)

(4, 1), (5, 2), (6, 3)

∴ Favorable outcomes = 6

∴ The probability for the difference of 3 between numbers on two dice:

=

=

**Therefore, the probability for difference between the numbers on the two dice being 3 is .**

**(ii) For product of numbers on two dice > 18, the possible outcomes are: **

(4, 5), (4, 6),

(5,4), (5,5), (5, 6)

(6,4),(6,5),(6,6)

∴ Number of favorable outcomes = 8

∴ The probability for product of numbers on two dice > 18:

=

=

**Therefore, the probability for product of the numbers on the two dice being more than 18 is .**

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